Rainsford has stayed on the deck to smoke. As he is standing there thinking of the conversation he and Whitney had just had about the island, he hears a gun shot. He leans over the side of the yacht so he can hear better and judge where the shot came from, and he drops his pipe. When he tries to catch it, he loses his balance and falls into the water.
At the beginning of the story, Rainsford and Whitney are on the deck of the ship. They are discussing various interpretations of hunting as well as the mysterious island off to the side of the ship. Whitney is the first character to decide to head to bed. Rainsford decides to stay on deck for a bit longer and have a smoke.
As Rainsford is relaxing in the stillness of the night, he hears three gunshots. Rainsford runs to the ship's railing and intently listens for further sounds. He hears nothing. That's when he decides to stand up on the railing. He figures that being higher will help him hear better. Unfortunately, as Rainsford is climbing up higher, his pipe gets knocked from his mouth. Instead of just letting the pipe fall away, Rainsford lunges after it, loses his balance, and falls overboard.
I'm not exactly sure where the single cause of Rainsford's misfortune is. If he didn't lunge after his pipe, then he wouldn't have lost his balance. If he didn't drop the pipe, he wouldn't have lunged after it. If he didn't climb the railing, his pipe wouldn't have been jarred lose. If he didn't hear the gunshots, then he wouldn't have climbed the railing. If he hadn't decided to stay on deck, he wouldn't have heard the gunshots. Of those possible choices, I feel the best choice is to say that climbing the railing is what causes Rainsford to fall overboard. That's the one thing that Rainsford has quite a bit of control over. He can't control whether or not he hears the gunshots, and lunging after the pipe was probably more reflex than conscious thought; however, he intentionally climbs that railing and put himself in a precarious position. It wasn't a smart decision, and he almost paid for it with his life.
"The Most Dangerous Game" opens with Rainsford on a yacht headed for the Amazon to hunt jaguars. It is nighttime and he describes the evening "like moist black velvet." This means that it is very dark outside, and probably dangerous if there is no lighting on the ship or the shore. After talking with a man named Whitney, he is left alone to smoke on deck while sitting on a "reclining steamer chair." Suddenly, Rainsford hears the sound of three gunshots. He gets up quickly from his seat to investigate and leans on the rail. With deep interest, he stands up on the rail thinking that some elevation will help him figure out where the shots are coming from. In the process, his pipe hits a rope and is knocked from his mouth, and he lunges to retrieve it. As a result, he loses his balance and ends up in the drink.