In the Two Towers movie, Frodo follows the lights and looks down into the marshes and sees a dead body. Then he "jumps" into the marsh.
Why does he do this?
Yeah, so this is the spectral corpse scene where was shot Frodo on a "dry for wet" set and all the ghouls and goblins were shot with high speed cameras on that "dry-for-wet". At Sony Imageworks they did all the shots in this sequence.
Basically the idea was that Peter gave us was that these could be real or they could just be these sort of tricks of the light so you didn't know whether there were actually these ghouls coming after you or whether Frodo was just sort of hallucinating this whole thing and so we made them a little particulate.
This doesn't happen in the book.
In the book it's actually Sam who falls, and he doesn't fall into the water:
Hurrying forward again, Sam tripped, catching his foot in some old root or tussock. He fell and came heavily on his hands, which sank deep into sticky ooze, so that his face was brought close to the surface of the dark mere. There was a faint hiss, a noisome smell went up, the lights flickered and danced and swirled. For a moment the water below him looked like some window, glazed with grimy glass, through which he was peering. Wrenching his hands out of the bog, he sprang back with a cry. 'There are dead things, dead faces in the water,' he said with horror. 'Dead faces!'
The history of the Dead Marshes is that were once the site of a battle before the Black Gate, which is described by Gollum:
There was a great battle long ago, yes, so they told him when Smeagol was young, when I was young before the Precious came. It was a great battle. Tall Men with long swords, and terrible Elves, and Orcses shrieking. They fought on the plain for days and months at the Black Gates.
This was presumably one of the battles of the Last Alliance, although I don't believe it's ever stated explicitly.
In the book the Marshes are described in a manner that implies they have some malign lingering influence as a result of this. Gollum describes the Marsh lights thus:
The tricksy lights. Candles of corpses, yes, yes. Don't you heed them! Don't look! Don't follow them!
And Frodo's description:
But I have seen them too. In the pools when the candles were lit. They lie in all the pools, pale faces, deep deep under the dark water. I saw them: grim faces and evil, and noble faces and sad. Many faces proud and fair, and weeds in their silver hair. But all foul, all rotting, all dead. A fell light is in them.
It's therefore evident that this malign influence lures travellers into the Marshes, and that Jackson took this scene and changed the emphasis for the movies; understandable in that it's a different medium, but it must be said that Frodo definitely doesn't jump in.
The marsh lights are known as Will-o'-the-wisp in folklore from Europe
In European folklore, these lights are believed to be spirits of the dead, fairies, or a variety of other supernatural beings which attempt to lead travelers to their demise.