The story opens with Rainsford's description as a hunter of wild game. He sees hunting as a sport between man and animal, a game between the hunted and the hunter. Rainsford doesn't think the animals feel anything about being hunted. It's a fun sport for people. When Rainsford meets Zaroff, this sport becomes a life-and-death game between two men. Rainsford, the hunter, becomes Rainsford, the hunted. It now becomes a game played for the highest stakes, and Zaroff has it rigged so he will win.
The author is making a statement about the violence and cruelty of hunting animals for sport. He puts a person in the place of the animal so we can imagine how the animal must feel being hunted. The title then reflects the "game" played between Zaroff and Rainsford, and it also reflects the author's theme that hunting isn't a game at all, and it means death for the animal.
The two meanings of the title "The Most Dangerous Game" are obvious to see as the story progresses. The beginning of the story tells of the conversation between Sanger Rainsford and Whitney, a sailor. They are talking about hunting animals and Whitney asks Rainsford about what the animal's feelings are when he is being hunted. Rainsford is quick to point out that you are either a hunter or huntee and luckily, they are both the hunters. But then the tables turn and Rainsford becomes the huntee so he can now relate to the fear of possible death when being hunted by Zaroff. He has to think and have the instincts that an animal would when being hunted.
Therefore "The Most Dangerous Game" refers to the game of hunting itself and game also refers to what is being hunted whether human or animal. In either situation they are both Dangerous Games.
Simply and succinctly, the double meaning of the title is that humans are the most dangerous "game animal" to hunt and that being the prey in a hunt is the most dangerous game to play. Rainsford is a renowned hunter, so hunting him will be a challenge for Zaroff, thus making the kill that much more of victory for him. Likewise, Rainsford must use his knowledge of the hunt and his tracking skills to avoid being killed, thus making it a dangerous game for him.