We cannot be 100% sure of what finally happens to General Zaroff, but the author makes very strong suggestions as to the outcome of the battle of wits between the two men. Evidently Richard Edward Connell decided that a violent ending would detract from the main drama in the story, which consists of one man being hunted through dense tropical vegetation by a skilled, sadistic hunter armed with a gun. Connell evidently wanted to keep it a battle of wits rather than a physical confrontation. So the end is swift and final. Everything that is needed to visualize what happens to Zaroff is contained in his dialogue when he discovers Sanger Rainsford waiting for him in his bedroom.
The general made one of his deepest bows. "I see," he said. "Splendid! One of us is to furnish a repast for the hounds. The other will sleep in this very excellent bed. On guard, Rainsford." . .
The term "On guard" is only used in dueling with swords.
He had never slept in a better bed, Rainsford decided.
Since Rainsford spends the night in Zaroff's bed, we understand that the two men had a sword fight in Zaroff's bedroom, Zaroff was mortally wounded, and then Rainsford got his final revenge by feeding the dead or dying man to his own savage, hungry hounds.