Why did the Americans win the war?

The colonies were far from Great Britain - this fostered a necessary independence on the part of the colonists and a benign neglect on the part of England. Because the Americans handled their affairs well, Britain was content to let them be for the most part.

The Americans knew better how to fight in that landscape of large tracts of wilderness. The European style of fighting was ill-suited to that landscape, and the Americans often engaged in guerilla-style warfare (thought dishonorable by the British)

France aided them significantly, particularly toward the end of the war, to such an extent that it contributed in a major way to the French financial crisis which precipitated the French Revolution only eight years after the American victory. Ironically the French supported the very ideas that later brought the French monarchy down.

They were well aware that their cause was right under existing English law. English law gave certain rights to Englishmen which Britain had violated (taxation without representation). Additionally, John Locke, the English philosoher, had articulated that man had "natural rights" which he called life, liberty and property. It was man's right and duty to overthrow their government if it abridged those rights (a right to revolution). His views were consistent with the existing rights given to Englishmen, and reinforced by the Enlightenment philosophes who articulated the ideals of equality and democracy (Jean-Jacques Rousseau), separation of powers (Baron Charles de Montesquieu), freedom of religion and speech (Voltaire). For the educated Englishment who were delegates of the continental congresses, the leaders of the revolution, these ideas were part of their standard education.