"The Most Dangerous Game" is a story about hunting. It specifically addresses hunting for sport or trophies rather than subsistence hunting for food. Perhaps the iconic upper-class version of hunting among the original audience would have been fox hunting, in which people entertain themselves by assembling large packs of dogs and groups of hunters to chase down a single terrified creature weighing eight or so pounds. The fox, in order to survive, hides, using small spaces inaccessible to its predators to evade capture and death.
Rainsford, like the fox, is being chased by hounds and has focused on surviving as prey. He realizes, though, that to win the game, he must cease acting like prey and become a predator. He chooses as an example of predator cats, who are stealthy, solitary hunters who choose their own place and time to pounce on unsuspecting prey.