Zaroff's "game" is the hunting and killing of human prey on his island.
Zaroff is a former Russian aristocrat who takes singular pleasure in hunting as a lifelong passion. Unfortunately for him, he has become bored with it, because he has hunted every animal worth hunting and found none of them to be a match for him. In his opinion, no animal can match a hunter with a good weapon and his wits about him. Therefore, he decides that humans are the only animal capable of giving him an interesting hunt, because they are rational. Yet Zaroff once again is disappointed because he finds that his usual lower-class prey, culled from wrecked ships, is not mentally astute enough to match him, and he remains unbeaten and rarely challenged.
Zaroff is excited for Rainsford to play this game because Rainsford is a famous hunter, and therefore Zaroff assumes that Rainsford will know how to make the game interesting for him, specifically by anticipating his plans, hiding his tracks and, apparently, fighting back.
Because Rainsford is an experienced hunter and will put up a challenge for him. Most of the other competitors were just ordinary people and didn't bother making traps, but relied on their feet and just ran.