That's a huge question. However, one could argue that it contributed directly to the downfall of the French monarchy and the rise of the French First Republic, with all that entailed. Three ways this could be backed are:
-The bankruptcy of the French government. Essentially, Louis spent the entire state treasury backing George Washington and fighting the British in an effort to get revenge for earlier French defeats. This had the effect of leading to massive financial and economic instability as the state became insolvent and currency became useless. The financial instability, and ensuing starvation and mass economic deprivation, led to the destabilization of power and the discrediting of the King.
-Exposure of the French to Republicanism. By this, I mean the anti-monarchist ideology rather than the modern-day GOP conservatism. Notable figures such as Count Lafayette cut their teeth on revolution in the New World, and then went back to France carrying the ideals of the Revolution with them. The midst of the financial crisis was a perfect time to deploy their experience and new ideas, which contributed to the French revolution's progress.
-The psychological impact of the British Empire's defeat was immense. The War of Independence was to England what Vietnam was to the US, except possibly worse. The idea that a rabble of upstart colonials could challenge and defeat a world empire did much to weaken the foundations of the Great Powers of Europe, and France in particular. In many ways, it fired up the spirits of the radical liberals who had been marginalized and were waiting for their chance. France, by far the most unstable of European nations, made an excellent launching-point.
In summary, the American Revolution had a massive ripple effect. Its importance in provoking the French Revolution was immense, and the latter event more than the former effectively changed politics, history, and philosophy on a profound level. At one point, some reporter asked a Chinese leader what he thought about the French Revolution. He replied that it was too early to tell. Its effects were incredible, and the Revolutionary War really set it off.