Rainsford is able to win the “game” by showing himself to be a stronger representative of the human race than his antagonist, General Zaroff. Zaroff had always believed in a warped kind of Darwinism, in which the strong have the absolute right to dominate and control the weak, and the weak, the so-called inferior species, go to the wall. Not surprisingly, Zaroff believed himself to be one of the strong.
However, it transpires that Zaroff couldn't have been more wrong. For it turns out that Rainsford was much stronger than him, as can be seen by the way that he outwits his pursuer during their “game”. Rainsford's superior survival skills, not to mention his outstanding skills as a hunter, enable him to gain the upper hand on the man hunting him down like an animal.
Rainsford may be a hunter himself, but by putting himself in the position of a hunted animal, he's able to turn the tables on Zaroff. This puts him at a distinct advantage to his pursuer, as Zaroff, with his obsession with the idea of the strong dominating the weak, is incapable of thinking like a hunted animal. Rainsford eventually drives home his advantage, with fatal consequences for the defeated Zaroff.