What causes Rainsford to change in the story?

Sanger Rainsford experiences a significant transformation and gains perspective after being hunted by the maniacal General Zaroff throughout Ship-Trap Island for three consecutive days. At the beginning of the story, Rainsford expresses his belief that the world is made up of two classes: "the hunters and the huntees." He believes that it is his inalienable right to exercise his strength and dominance over weaker beings. Rainsford also lacks sympathy for the animals he hunts and tells Whitney that the prey has no understanding of what is going on while they are being hunted.
Unfortunately, Rainsford falls off the yacht and must swim to General Zaroff's nearby island. During his first meal with the general, Rainsford is appalled to learn that Zaroff hunts humans throughout the island for sport. Rainsford ends up becoming Zaroff's prey and is forced to avoid the general and survive on the treacherous island for three consecutive days. As Rainsford is being hunted, he gains perspective and sympathy for the animals he hunts. He experiences the feelings of terror, fear, and anxiety as Zaroff closes in on him with his group of hunting dogs. By the end of the story, Rainsford illustrates his dramatic transformation by telling the general, "I am still a beast at bay...Get ready, General Zaroff." Rainsford's harrowing experience has made him more sympathetic to the animals he hunts and has expanded his perspective on the sport of hunting. Without becoming Zaroff's prey and narrowly surviving his game, Rainsford would have probably remained a callous, insensitive big-game hunter.

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.