The birth of the United States was actually the first practical application of the ideas of the Enlightenment in structuring a government from scratch. The founders of the country took many ideas directly from Enlightenment philosophers and built them into the foundation of the new democracy.
For starters, the Founding Fathers broke away from England because, they argued, the Crown was not protecting their natural rights to "life, liberty, and the pursuit of happiness." The idea of natural rights and a government's role in protecting them was espoused by John Locke. He argued that a government's power comes from the will of the people, not through the threat of force. The founders of the United States wanted to create a democracy that got its power from the popular will.
The structure of that government also came directly from the Enlightenment. The framers of the Constitution were concerned that too much power concentrated in one part of the government could lead to tyranny and abuses. That is why the government is structured largely along the lines espoused by Montesquieu, the great French philosopher who argued for the separation of powers in government. It is largely because of this Enlightenment philosopher that we have three branches of government (Executive, Judicial, Legislative) with checks and balances over each other.
The United States also guarantees certain rights to the people as espoused by other Enlightenment philosophers. Voltaire argued for freedom of speech and religion. Beccaria wanted governments to protect the rights of those accused of crimes. These are some of the earliest rights that people in the United States have and are enshrined in the Bill of Rights.
In short, many of the founders of the United States were students of the Englightenment. While they may have had many motives for waging a revolution against Great Britain, part of their reasons for it were to establish a democratic nation built on the principles of this era.