The two meanings are based on the word "Game". A game can be something played for enjoyment, or " animals under pursuit or taken in hunting; especially : wild animals hunted for sport or food" (see link below).
Zaroff tells Rainsford "No animal had a chance with me any more. That is no boast; it is a mathematical certainty. the animals had nothing but his legs and his instinct. Instinct is not match for reason."(pg 5) So Zaroff went in search of better game, and he found it. He says "it supplies me with the most exciting hunting in the world. No other hunting compares with it for an instant. Every day I hunt, and I never grow bored now, for I have a quarry with which I can match my wits." (pg 6) That quarry is the only one that can reason; that quarry is man. Since man can reason, he can match Zaroff's wits, and becomes the most dangerous game to hunt.
The second meaning is the game that is played. Zaroff has decided to hunt Rainsford. He tells him "You'll find this game worth playing".(pg8) Although Zaroff does not admit it at first, the game is to the death. That makes it the most dangerous game. Only one man survives.