hunting does change all that much throughout the story. He certainly understands, for the first time in his life, what it's like to be the hunted, but that doesn't mean that he suddenly starts questioning whether hunting is really such a worthwhile activity.

There's no sense that Rainsford will give up his favorite hobby because of his experiences on Ship-Trap Island. On the contrary, his turning of the tables on General Zaroff brings Rainsford considerable pleasure, so much so that he's able to sleep soundly in Zaroff's bed that night. This doesn't indicate someone whose conscience has been in any way disturbed—one presumes—by his killing of Zaroff or of turning the great white hunter into dog food.

Though there's little doubt that Rainsford has developed greater empathy for animals as a result of his experiences, there's every chance that he'll use that empathy to make him a more effective and more ruthless hunter in future.

Rainsford’s perspective on hunting does shift when he becomes the hunted instead of the hunter.  At the beginning of the story, Rainsford is on a boat headed for Rio and a hunting trip up the Amazon River. He hopes to have a good hunt for jaguars.  Whitney, his friend and fellow hunter, comments that hunting is a great sport for the hunter, but not for the jaguar. Rainsford responds,

“Don’t talk rot, Whitney…..You’re a big-game hunter, not a philosopher. Who cares how the jaguar feels.” (pg 1)

Whitney thinks maybe the jaguar cares how it feels.He thinks that they do understand one thing, fear.

“The fear of pain and the fear of death.” (pg 1)

Rainsford strongly disagrees.  He tells Whitney that the hot weather is making him soft.  He replies,

“The world is made up of two classes --- the hunters and the huntees.  Luckily, you and I are the hunters.” (pg 1)

When Rainsford meets Zaroff, he suddenly becomes the “huntee”. When Zaroff sends Rainsford out onto the island during their “game”, Rainsford becomes a series of animals.  First, he creates an intricate trail for Zaroff to follow recalling the, “dodges of the fox” (pg 7). When General Zaroff easily follows that trail, Rainsford tries to hide in a tree.

“Rainsford’s impulse was to hurl himself down like a panther.” (pg 8)

Zaroff smiles,and Rainsford realizes that the general is just playing with him,

“The Cossack was the cat; he was the mouse.  Then it was that Rainsford knew the full meaning of terror.” (pg 8)

Finally, when General Zaroff brings out his whole pack of dogs, Rainsford realizes how an animal feels when it is being hunted.

“The hounds raised their voices as they hit the fresh scent.  Rainsford knew now how an animal at bay feels.” (pg 9)

Rainsford continues to refer to himself as an animal even after General Zaroff tells him he has won the game. 

“I am still a beast at bay…” (pg 9)

Rainsford has changed his opinion.  He now knows how an animal feels when it is being hunted. 

My copy of the story is from the internet so the page numbers may not coincide with your copy.  

At the start of the hunt, Rainsford clearly was appalled by the idea that he would be the hunted. It was one thing to hunt animals, but the idea of hunting a human being was not something he had ever considered. However, as the tables turned and he used his skill to ultimately defeat Zaroff, if anything he has added another form of hunting to his repertoire. He does not feel that he is doing anything wrong by killing the hunter and usurping his "throne" (taking Zaroff's bed). This is clearly a change in attitude toward hunting, but not the change that one might have expected from the start.

I'm not so sure that Sanger Rainsford's love of hunting animals changed a great deal during the James Connell short story, "The Most Dangerous Game." Rainsford certainly comes to understand what it feels like to be hunted after his ordeal with General Zaroff. However, Rainsford seems more repulsed at Zaroff's idea of the ultimate hunt--stalking humans--than he does with hunting animals. The idea of hunting humans was sickening to Rainsford from the start, although he certainly used his repertoire of tricks to try and entrap Zaroff. At the end, when he settles into the wonderful bed and the well-earned sleep, he seems satisfied at the revenge that he has dealt the Russian. No doubt the human kill would not have tempted him to continue such "entertainment." Nor will he give up his quest for big game animals.

I do not see anything that would indicate that Rainsford's attitude towards hunting has changed over the course of the story.

You would think that being hunted himself might make him feel like hunting is not such a great thing after all, but we see no indication of that.

Instead, Rainsford uses tricks he has learned from hunting to stay alive.  He has no qualms about using those tricks to kill the dogs and the people following him.  And when he gets back to Zaroff's place, he doesn't feel bad about killing him and sleeping in his bed.  That implies that he is okay with the idea of hunting (and some people believe that he intends to take Zaroff's place and continue hunting the most dangerous game...)

It does seem that Rainsford has done an about face concerning his previous beliefs and statements concerning murder and the prey's understanding of fear and death. Rainsford had earlier told Whitney that the hunter's prey has "no understanding" of the fear of pain or death. But when Rainsford becomes the huntee of Zaroff, he soon feels all of the emotions that he had so easily dismissed. At the end of the story, when Rainsford surprises Zaroff in his bedroom, Zaroff honorably names Rainsford the winner of the hunt. Despite Zaroff's past acts of murdering his human victims, there is no reason to believe that he will not honor the rules of the game. Instead, Rainsford wants to continue the hunt, and this time he apparently kills the Russian. Whether Rainsford's days on the run has made him envious of switching places with Zaroff, or whether his act is simply one of revenge, Rainsford has taken to murder, and he seems content with the result.

He had never slept in a better bed, Rainsford decided.

Although dead tired from his ordeal, the final line seems to show that Rainsford has also enjoyed this final hunt, and one can only wonder what prey he will next choose to stalk.

One of the ironies of Connell's famous story is that in the exposition of "The Most Dangerous Game," Rainsford and his friend Whitney, who are on the ship in the Caribbean night, argue about the prey that they will soon hunt. Whitney contends that the jaguar possesses an understanding of pain and fear; Rainsford disagrees,

"Nonsense....This hot weather is making you soft, Whitney. Be a realist. The world is made up of two classes--the hunters and the huntees. Luckily, you and I are hunters."

Rainsford has also remarked, "Who cares how a jaguar feels?" But, after his harrowing experience as a "beast at bay" as he calls himself when he and the general come vis-à-vis in the Zaroff's bedroom, Rainsford has probably changed his attitude about hunting. While he yet prefers to be the hunter--he revels in his victory as he sleeps in Zaroff's bed, having defeated this predator--surely, Rainsford must now consider the feelings of his prey since since having had this experience himself. It is, therefore, most likely that before he shoots whatever he hunts in the future, he may pause for a split second and recall the gripping fear of the "beast at bay" that he himself has known. 

No.  I do not think that the ordeal with Zaroff changes Rainsford's mind about hunting.  I think he still enjoys hunting.  I think he still is fairly cold.  And I still think he finds hunting humans distasteful.  

Rainsford is a world renowned hunter.  I don't think his ordeal changes anything, especially knowing that he slept quite soundly after killing Zaroff.  Early in the story the reader is introduced to the fact that Rainsford is good at hunting and has little sympathy for his prey.  

“…We should have some good hunting up the Amazon. Great sport, hunting."

Rainsford: "The best sport in the world."

"For the hunter," amended Whitney. "Not for the jaguar."

[…]

"Bah! They've no understanding."

"Even so, I rather think they understand one thing--fear. The fear of pain and the fear of death." 

Rainsford doesn't even believe that the animals he hunts have fear.  They have no understanding of what is happening.  I think Zaroff feels the same way, which is why he enjoys hunting the only animal that can reason -- humans.  Rainsford is appalled and intrigued all at the same time, but ultimately doesn't agree to hunt a human.  The downside of that decision is that he becomes the hunted.  

I've read an analysis or two that say Rainsford enjoyed killing Zaroff and is likely to begin and enjoy hunting humans too.  Basically Rainsford will continue Zaroff's sadistic tendencies, and that's why Rainsford slept so well.  I disagree.  I think Rainsford slept so well because he knew that his life was no longer in danger.  He killed Zaroff out of self defense, and he can now finally relax after his three days of not doing much sleeping.  It's even conceivably possible that Rainsford has become even more immune to any feelings that his prey might have and that's why he slept great.   It's my guess that he goes back to hunting with an even cooler/colder attitude toward his prey than before.  

There is some evidence that Rainsford does change his mind about hunting in "The Most Dangerous Game" near the middle of the story.

In the exposition as Sanger Rainsford and his friend Whitney travel to Rio de Janeiro to pick up guns with which they will hunt jaguars, the two men discuss the dynamics of the hunt. Whitney remarks that the hunt will be no fun for the jaguar, but Rainsford interjects, "Who cares how a jaguar feels?" Still, Whitney speculates that the jaguars understand fear when they become man's prey--"The fear of pain and the fear of death." But Rainsford dismisses this opinion: "Bah! They've no understanding."

After Whitney retires for the night, Rainsford hears gunshots and he tries to see from where it has come by jumping up on the rail. However, he falls from the yacht into the dark waters of the sea on a moonless night. When he surfaces, he tries to get the attention of someone on the yacht, but the ship keeps going. After swimming a great distance, Rainsford finds the shore. On the next day, he discovers a chateau on the island and climbs toward it. Unfortunately for Rainsford, this beautiful place is inhabited by General Zaroff, a sadistic and experienced hunter who finds that he is only excited by hunting humans. To his shock, Rainsford learns that he is to be hunted the next day in this "most dangerous game" of man versus man. For the first time, Rainsford is the prey rather than the predator.

Soon thereafter, he begins to understand some of what his friend Whitney has said about the huntee's experience of fear. One could make the case that his own feelings of fear as he is hunted are strong enough to change his mind about hunting; he realizes how it feels to be the prey. It's important to note, however, that Rainsford never explicitly states that he has changed his mind about hunting.

On the second day as he is hunted, Rainsford is terrified; he has climbed a tree in the hope that the pursuing Zaroff will not find him. But the general, who possesses uncanny powers, has followed his trail in the dark. When he sees the general look up and smile, "Then it was that Rainsford knew the full meaning of terror."

After he slides down from the tree, Rainsford pulls out his knife and sets to work on a Malay man-catcher, but when Zaroff returns, the general is able to dodge the trap, injuring only his shoulder. Rainsford feels "fear again ripping his heart..." because he knows that the predator Zaroff will return.

Toward the end of the story, Rainsford will explicitly identify himself as "a beast at bay"—suggesting he has recognized the fear he feels in the middle of the story is akin to what the animals he hunts may feel. Though he never says he's changed his mind about the merits of hunting, one could use this evidence to build an argument for the idea that he won't go on to inflict the fear he has felt on other creatures.

"The Most Dangerous Game" is an example of the slick, commercial short fiction that used to be published in magazines but has pretty much gone out of fashion because the people who read such escapist fiction now spend their leisure time watching television or playing video games instead. Editors of the magazines that published adventure-escapist stories had one rule in common. They thought that the major character should change as the result of his harrowing experiences. If he didn't change, then the experience couldn't have been very important or very hazardous. So commercial writers would take care to establish that the viewpoint character had changed by the end of the story, even though he might not appear to be much different to the reader. Sometimes the writer would even wind up with dialogue such as this:

"You've changed."

"Have I? Yes, you're right. I guess I have."

Is it really true that a person's character would be changed if he or she went through a really traumatic experience?

"The Most Dangerous Game" is just a superior work of slick fiction. It was published in onw the better class magazines which paid more money. They were printed on "slick" paper, as opposed to the "pulp magazines" which were printed on cheap paper and paid as little as one cent a word, whereas the "slicks" would pay around ten cents a word, and more to authors whose names had value in selling copies of the magazines. Most of the slick magazines that published short fiction have gone out of business or else converted to articles. Most of the pulp magazines have disappeared. There used to be whole rows of pulp magazines at drug stores, grocery stores, and news stands. There were westerns, mysteries, romances, true detectives, science-ficition, and others. Television changed all that. "The Most Dangerous Game" would definitely hCW gone to a slick magazine because of the good quality of the writing and the intriguing idea, but it is not serious literature. Readers identify with the hero because they keep wondering what they would do themselves if they were placed in the same perilous situation.

Certainly, after his experiences on Ship-Trap Island Sanger Rainsford has gained new perspectives about hunting.

• In the exposition of the story while he and Whitney talk in anticipation of their hunting of the jaguar, Whitney remarks that as prey the beasts know the fear of pain and the fear of death, but Rainsford dismisses this observation as "nonsense."
• After he falls overboard and washes up on Ship-Trap Island and is taken to the chateau of Gerneral Zaroff, Rainsford is appalled when his host explains what he means by "more dangerous game."
• While he is involved in this "more dangerous game" as the prey, Rainsford learns what it is to be "a beast at prey" as he hides upon the limb of a tree; thus, he changes his attitude expresses earlier as "nonsense" and knows that those fears Whitney has mentioned are real:

Then it was that Rainsford knew the full meaning of terror....He lived a year in a moment.

• After leaping into the sea, Rainsford returns to the general's chateau when, as the general himself has reflected before retiring for the night, "his quarry escaped him."

This action of returning to confront Zaroff and his reaction after he kills the general--"He had never slept in a better bed"--indicate that Rainsford has changed. For, having learned how prey feel and having enjoyed the "most dangerous game," of hunting and killing Zaroff, Sanger Rainsford has, indeed, changed his mind about hunting.