The simple answer is, very significant. The American war is frequently understood as part of the 'second hundred years' war' between Britain (England until 1707) and France. Effectively, the American patriots switched sides.

Basically, the British had won the Seven Years' War, but at the cost of near-bankruptcy. They were left with the problem of defending the North American colonies, but with the additional threats of Indians who had allied with France, and the annexed population of New France (Quebec). The various British responses - attempts to raise money from the colonies, prevent settler antagonisation of the Indians, and garrison troops - were among the major causes of offence that led to the Declaration of Independence.

The Declaration itself, and the diplomatic overtures by the Continental Congress to France, show how important the rebels thought French involvement was. France was able to field roughly 12 000 troops in the North American theatre, as well as provide naval support, but also threaten the United Kingdom with invasion, which was critical in fostering domestic opinion in the UK that the war was being mismanaged and should be ended.

In reference to me, I'd like to clarify. The defeat of the French at the hands of the Prussian and the British during the Seven Years War created an identity crisis in French military circles. So, they figured that something needed to be done and there was a lot of discussion about how war should be fought, even some suggesting popular armies made of the people. However, most of these ideas came at the tail end of the American Revolution and we're being tested in the last decade before the French Revolution. So, in respect to military theory, it would be very little but that doesn't mean that the French didn't have a hand in training the Americans although I don't know to what extent.