One of the major themes of Steinbeck's novella Of Mice and Men is the importance of friendship, especially for the working men on the ranch who have mostly been marginalized and isolated. At one point Slim points out that it's funny to see two men travel together as George and Lennie are doing. He asserts, in chapter three, that the laborers usually travel alone and don't care about each other:

"Hardly none of the guys ever travel together. I hardly never seen two guys travel together. You know how the hands are, they just come in and get their bunk and work a month, and then they quit and go out alone. Never seem to give a damn about nobody."

Fortunately for George and Lennie, they have each other and their friendship appears to be quite strong. George admits that Lennie is a "nuisance" but he continues to travel with him, and they form an important bond. Likewise, Candy sees the old dog that he raised from a puppy as his close friend, and, as with George and Lennie, they have a meaningful bond. It is why he balks at the thought of killing the old dog, despite the dog's infirmities. Candy's offer to join George and Lennie in their quest for their own piece of land is compensation for the loss of his only friend.

Even before the old dog is taken out and shot by Carlson, Steinbeck weaves in his ideas about friendship when he brings up the letter in the pulp magazine which Whit shows to Slim. What is seemingly a meaningless scene in the book is really a commentary on how important friendship is for these working men. Whit had worked with the man, Bill Tenner, who wrote the letter, and they had become friends. Whit says, "Bill was a hell of a nice fella." This display of affection is juxtaposed with Carlson's selfish demands that Candy's dog must be put out of its misery. Carlson's lack of empathy for friendship is again shown in the book's final scene when he fails to understand George's grief over killing Lennie. He remarks ignorantly as George and Slim walk away from him, "Now what the hell ya suppose is eatin' them two guys?"

Many people become attached to pets and don't want to have them "put to sleep" when they are old and sick. The whole business about Candy's dog was put into the story to establish that there was a gun available, so that George could steal it and use it later to kill Lennie. Carlson explains where to point the gun so that the dog's death will be painless, and so George knows where to point it when he shoots Lennie. Furthermore, Carlson cleans the Luger carefully after using it, and George is able to see exactly how the German automatic pistol works and where it is kept under Carlson's bunk. Carlson is a middle-aged man. We assume without being told that he served in the army in Europe during World War I and brought the Luger back as a souvenir. Steinbeck wanted a handgun that was readily recognizable, so that when George takes it out of his jacket to shoot Lennie, the reader (and eventually the theater audience when the novel is adapted into a stage play) will understand that it is Carlson's gun. A German Luger is a very distinctive-looking weapon. Steinbeck knew that every scene in a work of fiction should be dramatic. Drama involves conflict. He devised a conflict between Carlson and Candy over killing the dog in order to make the scene more interesting. The conflict is: Carlson is bothered by the old dog's smell and wants to shoot it; but Candy is attached to the dog and doesn't want to let Carlson shoot it. The conflict is resolved when Candy yields to pressure and the dog is shot. Notice the amount of explanation Steinbeck has given to the way Carlson unloads, cleans, and reloads his Luger after shooting the dog. The gun is what in Hollywood is called a "plant." A weapon is shown in advance, so that the audience won't wonder where it came from when it is used later in the story. If George had pulled a gun out of his pocket when he joined Lennie at the river, the reader/viewer would wonder where he got it if he didn't recognize it as Carlson's Luger. George intended to kill Lennie from the time he left the ranch. His possession of the stolen Luger proves his intent. The men thought Lennie had taken it when he fled. This was convenient for George, mainly because he could let everybody assume that he had taken it away from Lennie and shot him in self-defense, using this as an alibi to avoid prosecution for what was actually first-degree, premeditated murder.

The most relevant answer to your question is that Candy does not want to kill his dog because it is an extension of him. What I mean by this is that since the dog has no "real" use left Carlson wants to kill it; we, as readers, can see that Candy is already thinking that he himself does not have much more "useful" time left. This also goes back to the theme of "usefulness" in the text. As a secondary answer this event also points to the idea that there is to sentimentality on the ranch: once something has ceased to serve a purpose it is discarded. Candy’s dog has been his only companion for years and years and yet when it is recognized as not being able to serve a purpose or there is something else that can do a better job (i.e. Slim’s puppies) then it is discarded. Lastly, this foreshadows the end of the book and Lennie’s eventual death at the hand of George.

Quite simply, Candy loves the very old dog which he had kept since the dog was a pup. Candy's dog is his companion, having kept him company and given him affection for many years. When Carlson first suggested that it was time to shoot his dog, Candy replied softly, "No . . . No, I couldn' do that. I had 'im too long." Carlson persists. Candy resists, trying to come up with an argument to stop Carlson: "Maybe it'd hurt him . . . I don't mind takin' care of him." During the discussion, Candy is afraid of what's coming. Carlson presses the point. When Slim won't intercede on Candy's behalf, Candy gives up and gives in. Carlson takes the dog outside, and after a time, a single shot is heard. Lying on his bunk, staring at the ceiling, Candy then turns over and faces the wall. He loved his dog and is now alone without him.

On the simplest level, Candy doesn't want to kill his dog because he loves it and remembers lots of great times they had together.

On a more symbolic level, Candy knows that shooting the dog would be like shooting himself. (His dog is old and weak; he is disabled. To agree to get rid of one would be like getting rid of the other.)