The island does have an evil reputation. It is introduced as Ship-trap Island right from the beginning of the story. Here is what Whitney says:
"OFF THERE to the right--somewhere--is a large island," said Whitney." It's rather a mystery--" "What island is it?" Rainsford asked.
"The old charts call it `Ship-Trap Island,"' Whitney replied." A suggestive name, isn't it? Sailors have a curious dread of the place. I don't know why. Some superstition--"
The name suggests that it is place where ships are wrecked. This also entails, more ominously, those who are on these ships. In view of this, the reader gets the feeling that the island will trap people for ill. At first, the reader cannot guess what will happen for sure, but as the story progresses, it become clear that the island will ensnare people and make them play a "game."
The game is a game of hunting, where the people are the prey. General Zaroff created this game, because he was tired of hunting animals. According to Zaroff, animals presented little challenge, because they could not think. So, he started to hunt a rational animal, man.