How did america win the revolutionary war?

One thing going against the British was that there wasn't any center of gravity in the Colonies - after major cities like Charleston, New York, Boston, or Philadelphia were captured, the War kept on since the Americans were not centralized at that time. The size of the colonies played a role in the British defeat as well, as the British had to simultaneously fight a war and occupy the colonies to suppress the rebellion. They could easily fight the war, but not garrison troops, as doing so meant that those trops could not be readily sent into combat without allowing revolutionary sentiment to creep back into a previously occupied area. The British couldn't conduct suppression operations the same way they did in places like Ireland and Scotland, because doing so would mean that the British would lose the support of the Loyalists that they so desparately needed to maintain to ensure that they had any chance at all of winning the War, thus they were limited in how brutal they could be and whether they could employ slaves and American Indians to fight with them (keep in mind the greatest concentration of Loyalists existed in the American South) - especially after the hiring of Hessian mercenaries proved as controversial as it did. 

In the end the combination of American tactics, European aid/intervention, and the limitations imposed on the British by fighting a war from across the ocean were the major reasons why the Colonies won their war for independence.

They said that the world was flat