How were Bilbo and the dwarfs able to cross the river in The Hobbit?

The travelers use a rope to catch a boat from the other side of the river.

The forest of Mirkwood is a dangerous place.  You can lose your head easily.  Beorn warned the company to stay out of the water because it would bewitch them.  When they came to a river, they could not ford it as they had the others.

This was their state when one day they found their path blocked by a running water. It flowed fast and strong but not very wide right across the way, and it was black, or looked it in the gloom. (Ch. 8)

Bilbo suggests that the river is about twelve yards wide, although Thorin doesn’t believe him.  He thinks it is thirty yards.  Bilbo comes up with a plan to get the boat they see across the river.

"Can any of you throw a rope?"

"What's the good of that? The boat is sure to be tied up, even if we could hook it, which I doubt."

"I don't believe it is tied," said Bilbo, "though of course I can't be sure in this light … (Ch. 8)

The plan works, although it takes a few tries.  Fili is finally able to hook the boat, and the others help to reel it in.  Bilbo had not thought the boat was tied up, but it turns out it was.  Still, with their combined strength they are able to get it.

They break up into groups since they will not all fit in the boat at once.  Bombur complains about being last, saying "I'm always last and I don't like it," but Thorin tells him it is because he’s fat.  Unfortunately, Bombur falls into the river and the boat disappears.  There is no way to get back.  The dwarves try to shoot at some deer, but they are not successful and they waste all of their arrows.

The incident is just one of several difficulties the travelers face on their way to the Lonely Mountain.  They cross many rivers, and most of them are easy to ford.  This river, however, is dangerous.  The travelers have to be careful not just of the speed and width of the river, but of the water inside it and its effects.