In the Most Dangerous Game, Rainford agreed to be the hunted because he did not have a choice after the despicable General Zaroff got him. In this story, Rainford himself is a (human) hunter and being the prey made him excited more than dwelling of his odds of being killed.
He knows the ways of a hunter and that gives him the edge in saving his life against Zaroff. As a person, Rainford loves a challenge and he considered being the hunted the greatest challenge he would experience. He has knowledge of the game, he knows his hunter, and he wants to discover his strengths that are yet to unleash.