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.