As Rainsford tries to elude General Zaroff, he uses his knowledge of woodcraft and hunting to lay false trails. He compares his ploy to that of a fox, hunted by dogs, which will sometimes cross streams, double back, and pass through thorny bushes to avoid capture. After he spends hours laying the false trail, he realizes that he needs to rest.

He knew it would be insane to blunder on through the dark, even if he had the strength. His need for rest was imperative and he thought, "I have played the fox, now I must play the cat of the fable."
(Connell, "The Most Dangerous Game," fiction.eserver.org)

The reference is to Aesop's fable of the Fox and the Cat, where the Fox declares that it has many different skills to escape hounds, while the Cat has only one. When the dogs come, the Cat climbs a tree, while the Fox is caught and killed. Rainsford, climbing a tree, seeks to escape from both Zaroff and his dogs, hiding off the ground so that his trail seems to simply stop.

In "The Most Dangerous Game," what does Rainsford mean by "play the cat of the fable"?

"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.

The fox is the animal being chased.  Rainsford has been chased all across the island.  He is tired of being the "fox."  He is tired of being hunted. He says he must now become the cat in the fable.  The cat is the hunter and Rainsford deciders to begin to think as a hunter instead of being hunted.  His role is reversed.