The Enlightenment influenced America's belief that we, the people, have a right to form our own government.  The American democracy's arrangement  of a separation of powers (checks and balances) also came from Enlightenment thought.  There were other influences too, but those are two big ideas that came from the Enlightenment and were implemented in the USA.
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Ideas about the people's sovereignty came from thinkers like John Locke of England and Jean-Jacques Rousseau of France.   Both of them were writing philosophy about how the people should be the ultimate authority  in a state -- that a government gets its power from the people and needs to serve the interests of the people.  That idea was referred to as the "social contract."  Rousseau famously wrote a book with that title -- The Social Contract -- published in 1762.  John Locke had published Two Treatises on Civil Government in 1690.  Locke repudiated the views of divine right monarchy in his First Treatise on Civil Government.  In his Second Treatise,  Locke argued for the rights of the people to create their own governments according to their own desires and for the sake of protecting their own life, liberty, and property.  Locke always favored the people remaining in charge, and asserted that the people have the power to change their government and remove government leaders if the government is not properly serving the needs and well-being of the people.  The "Separation of Powers" principle was an idea embedded into the plans for American government by our founding fathers, based on their reading of Enlightenment political theory.  The terminology "separation of powers" was introduced by Charles-Louis de Secondat, the Baron of Montesquieu.  (Usually he's referred to as just "Montesquieu.")  He wrote an important work of political theory called The Spirit of the Laws, published in 1748.  Within his treatment of how governments will function best, Montesquieu argued that executive, legislative, and judicial functions of government ought to be divided between parts of the government, so that no one person or division of the government can infringe on the overall rights of others in the government or of the members of the society overall.
