When he attempts to catch his pipe as it falls from his mouth, Sanger Rainsford loses his balance and falls in the Caribbean Sea. Rainsford's plunge overboard, which occurs on a pitch black night, goes unnoticed by others on the yacht, so he is forced to attempt to swim to the only land nearby, Ship-Trap Island.
Although Rainsford nearly drowns in the process, he does manage to swim to the island. Rainsford knows that there are other people on the island, because he heard gunshots coming from that location shortly before he fell into the sea. Rainsford follows the shoreline of the island, comes across footprints, and continues in the direction in which the boot tracks point.
After some time, Rainsford comes to
...one enormous building--a lofty structure with pointy towers plunging upward into the gloom. His eyes made out the shadowy outlines of a palatial chateau; it was set on a high bluff, and on three sides of it cliffs dived down to where the sea licked greedy lips in the shadows.
Rainsford knocks on the door, which is answered by a huge, bearded man. Behind that man, an "erect, slender man in evening clothes" comes down a set of stairs and introduces himself as General Zaroff. Zaroff is a strange-looking man whose
...hair was a vivid white; but his eyebrows and pointed military moustache were as black as the night from which Rainsford had come. His eyes, too, were black and very bright. He had high cheek bones, a sharp-cut nose, a spare, dark face, the face of a man used to giving orders, the face of an aristocrat.
Zaroff has read a book written by Rainsford and recognizes him.
Rainsford is supposed to be a man of the world, a great adventurer, a fearless hunter, and so forth. And yet he looks like an idiot when he allows himself to fall off a yacht in what is known to be a shark-infested ocean. Evidently the author Richard Edward Connell couldn't think of a better way to have his hero end up on an island where a madman hunts human beings for sport. Rainsford smokes a pipe because stalwart heroes in adventure yarns always smoked pipes in those days. Maybe a pipe makes a man looks wise, and the author must have wanted Rainsford to be smart as well as courageous. But why should he drop his pipe in the first place? And why should he be fool enough to fall overboard while trying to catch it? In The Island of Dr. Moreau by H. G. Wells, the viewpoint narrator Edward Prendick lands on Dr. Moreau's island because he was shipwrecked. That is far more plausible.
Rainsford falls off of his steam ship and washes up on Ship-Trap Island. Once there, he wanders through the jungle until he comes across General Zaroff's chateau where he meets Ivan and then General Zaroff.