The principles of American democracy are founded to a large degree upon the tradition of English liberty ultimately going back to the Magna Carta. English democracy evolved over a period of hundreds of years, and we can look at the English Civil War of the 1600's and the "Glorious Revolution" of 1688 as great forward bounds in this evolution. In declaring their independence from Britain in 1776, Americans were in fact asserting what they believed were their rights as Englishmen, which they saw as having been violated by George III and his administration. And in doing so they were extending those rights, drawing on the ideas of philosophers such as John Locke, whose writings are in some sense a codification of the principles of the English Constitution.
What is the connection between these facts and the overall European Enlightenment of the 1700's ? First, it's obvious that Enlightenment philosophers such as Voltaire were influenced and inspired by the (relative) political freedom they knew existed in Britain in comparison with the mostly absolutist forms of government in the countries of Continental Europe. Second, the Enlightenment was a secular movement in which the adherence to religion, and its corollary notion that kings ruled by divine right, were rejected. American democracy is based on the separation of church and state. This, of course, is a change from the English tradition in which the Church of England is linked to the government.
The United States was thus created from a chain of influences beginning with the English Constitution and followed by philosphers such as Locke, Hume, Voltaire and Rousseau, who extended the general principles of liberty, secularism, and the natural and equal rights of human beings. This may be a simplified explanation of how the Enlightenment inspired the American Revolution and ultimately the U.S. Constitution, but it can serve as the basis for further research you might wish to undertake on the subject.