The cause of Rainsford's change is that he goes from being the hunter to being the hunted.
At the beginning of the story, Rainsford shows zero care for the animals that he hunts. He feels that the animals are prey and nothing more; they are not thinking and emotional creatures. They operate on nothing more than instinct; therefore, they do not feel fear.
"You're a big-game hunter, not a philosopher. Who cares how a jaguar feels?"
"Perhaps the jaguar does," observed Whitney.
"Bah! They've no understanding."
"Even so, I rather think they understand one thing—fear. The fear of pain and the fear of death."
"Nonsense," laughed Rainsford.
All that changes after being shipwrecked on General Zaroff's island. Zaroff hunts humans, and he makes Rainsford his prey. Rainsford is appalled at the scenario; however, he has little choice but to run and fight for his survival. Once the hunting begins, Rainsford gets to see, feel, and understand the other side of the hunt. He gains a new respect for the animals that he hunts because he now understands what it feels like to be pursued by a relentless killer that has zero care for how his prey feels. Because he is being hunted by Zaroff, Rainsford now fully understands and empathizes with the animals that he once hunted. That is why the following quote from Rainsford is so important.
"I am still a beast at bay," he said, in a low, hoarse voice.
He does not say that he's a man seeking vengeance or even self-defense. Rainsford labels himself "a beast." He sees himself as an animal, and he is no longer thinking like a human. He's been cornered into a situation in which he absolutely must kill or be killed. That's what happens when an animal is cornered; it will fight harder than it ever has before with no concern for the other aggressor. That is now Rainsford. Gone is the opinion that killing another human being is morally suspect. That change in Rainsford would not have happened without his island experience.