Rainsford's fall from the yacht is caused by his lunging for his pipe, which is knocked from his mouth as it strikes a rope after he leaps upon a rail. 

Sanger Rainsford and his friend Whitney travel on a yacht through a moonless night as they are on their way to hunt jaguars "up the Amazon." During an evening conversation, Rainsford and Whitney disagree on their perspectives about hunting these jaguars. Rainsford has no concern whatsoever for the prey that he hunts, while Whitney speculates about the fear that such prey must feel. After this conversation (which acts as foreshadowing), Whitney retires for the night while Rainsford stays on deck since he is not sleepy. "I'm going to smoke another pipe up on the afterdeck," he tells Whitney.

As he relaxes with his "favorite brier," Rainsford is startled by "an abrupt sound." He hears this sound repeated, a sound that he recognizes immediately as the firing of a gun. Then, in his effort to see through the darkness of a moonless night, Rainsford springs quickly onto the ship's rail. As he balances himself, his pipe is knocked from his mouth by a rope. Quickly, Rainsford reaches out for it, causing himself to lose his balance and fall into the "blood-warm waters of the Caribbean Sea." He cries out when he resurfaces from the sea's depths, but no one hears him. Even when he tries to swim closer to the receding yacht, Rainsford knows his efforts are futile. Reaching for the pipe has been a costly error, as he expended a lot of energy doing so but now must swim to shore.

In "The Most Dangerous Game," what exactly causes Rainsford to fall off the yacht?

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.

In "The Most Dangerous Game," what exactly causes Rainsford to fall off the yacht?

"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.

In "The Most Dangerous Game," what exactly causes Rainsford to fall off the yacht?

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.

What events led Rainsford to fall into the sea in "The Most Dangerous Game"?

Rainsford has a conversation with Whitney about the island, hears a gunshot, and then falls off the yacht.

Rainsford and Whitney are hunters on a yacht in the Caribbean Sea. They discuss the spooky reputation of the island they are passing by.

"The old charts call it `Ship-Trap Island,"' Whitney replied." A suggestive name, isn't it? Sailors have a curious dread of the place. I don't know why. Some superstition--"

"Can't see it," remarked Rainsford, trying to peer through the dank tropical night that was palpable as it pressed its thick warm blackness in upon the yacht.

They conversation soon turns to hunting, and whether or not animals have real feelings. After this conversation, Rainsford gets pensive. He goes out on the deck to smoke, saying he is not sleepy, and watches the water.

Rainsford gets curious when he hears a gunshot on the island.

An abrupt sound startled him. Off to the right he heard it, and his ears, expert in such matters, could not be mistaken. Again he heard the sound, and again. Somewhere, off in the blackness, someone had fired a gun three times.

This is actually foreshadowing. At this time, Rainsford has no idea what is happening on this island. He does not know that there is a crazy man hunting human beings there.  He just hears a gunshot and gets curious. However, it is the gunshot that makes him fall overboard.

Rainsford is a skilled hunter, and a survivalist. He is able to swim ashore, and even though it is difficult, he survives the trip to the island. He has no idea that the swim was the easiest part! Once he is there, he finds a bullet cartridge that is way too small to hunt an animal large enough to have made the thrashing in the underbrush. Rainsford is confused, and curious. Things are definitely strange on this island.