This quote occurs in the opening of the story, after Whitney explains the legend of Ship-Trap Island to Rainsford. Rainsford is a well-known hunter and so is attuned to the sounds of gunfire; he does not expect to hear any on the ocean, even nearby an island. The sound of the shots is the direct cause of Rainsford falling overboard, and he orients himself to the island by a final shot:
Off to the right he heard it, and his ears, expert in such matters, could not be mistaken... Somewhere, off in the blackness, someone had fired a gun three times.
"I suppose the first three shots I heard was when the hunter flushed his quarry and wounded it. The last shot was when he trailed it here and finished it."
(Connell, "The Most Dangerous Game," classicreader.com)
As later events show, the three shots were the culmination of one of General Zaroff's manhunts on the island. To create greater tension, Zaroff hunts alone with only a small-caliber pistol. Since he is also a great hunter, Zaroff inevitably finds his prey and kills it with his pistol; Rainsford correctly infers that the shots he heard was from a hunter, although he does not yet know which game Zaroff is hunting. Flushing game with shots -- to wound or only to scare -- is a classic hunting maneuver, and Zaroff knew that once he drives a man to ground, it only takes one or two more shots to finish the job. Rainsford, of course, becomes the prey later on and is barely able to keep ahead of Zaroff.