How did the ideas of the enlightenment philosophers most influence popular revolutions in france and in the united states

Enlightenment thinkers promoted the idea of the rights of citizens and the people's authority to create--and to change--their own governments.  The works of Enlightenment philosophers such as John Locke, Baron de Montesquieu, and Jean-Jacques Rousseau were read by leaders of the revolution movements in America and in France.  The American and French Revolutions sought to put those Enlightenment ideas into practice in creating new governments based on liberty and justice for all, and on the rights of man and of citizens.

As an example of one Enlightenment philosopher's political thoughts that influenced the revolutions, let's look at John Locke.  According to Locke's  view, a government's power to govern comes from the consent of the people themselves -- those who are to be governed.  This was a change from the previous ideas of "divine right monarchy" -- that a king ruled because God appointed him to be the ruler.  Locke repudiated the views of divine right monarchy in his First Treatise on Civil Government.  In his Second Treatise on Civil Government, 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.

The American founding fathers read Locke (as well as other Enlightenment writers like Montesquieu and Rousseau).  The American Revolution (1775-1783) was inspired by these ideas.

The French Revolution (1789-1799) also took cues from the political philosophy of John Locke and Montesquieu--and especially also Rousseau--overthrowing the unwanted monarchy of Louis XVI.