Your question goes to the heart of the story and how General Zaroff has "advanced" beyond animal game to something more interesting. Note in particular the conversation between Rainsford and Zaroff when Rainsford notices the Cape Buffalo head hung up. Rainsford comments that he thought this animal was the most dangerous of all big game, but Zaroff replies, with "his curious red-lipped smile":
"No. You are wrong, sir. The Cape buffalo is not the most dangerous big game." He sipped his wine. "Here in my preserve on this island," he said in the same slow tone, "I hunt more dangerous game."
Note Zaroff's further descriptions of how he has had to "stock" the island, and how there is "no thrill left in tigers." Zaroff says that now he lives "for danger". Later, we see how Zaroff has become so good at hunting that hunting for animals had lost its attraction for him. So he thought about what animal would give him the pleasure of the hunt that he had lost:
"I wanted the ideal animal to hunt," explained the general. "So I said: 'What are the attributes of an ideal quarry?' And the answer was, of course: 'It must have courage, cunning, and, above all, it must be able to reason.'"
Of course, as Rainsford is able to deduce, there is only one kind of animal that has these characteristics: the human. This is how Zaroff has made hunting interesting for himself again - he hunts humans. This introduces the breathtaking hunt between Zaroff and Rainsford that dominates the rest of the story.
When General Zaroff became bored with hunting animals, he did two things. First, he bought himself the island that the story mostly takes place on. Second, he started to hunt people instead of animals.
Having the island allowed him to have a place where he could do that. If he had been living on any place other than a little island that he owned, he would have had to deal with other people. There might have been a government that would investigate what he was doing. But living on his own island, he does not have to worry about getting caught hunting and killing human beings.
Hunting people was more interesting than hunting animals because people are smarter and more able to give him a challenge than animals were.