How did Enlightenment ideas encourage people from the American colonies and France to change their governments?

What both the American and French Revolutions have in common is the direct challenge to absolute monarchy -- the rejection of the Hobbesian idea that human nature requires a strong monarch, or a Leviathan -- to control and properly direct the worst impulses in a populace. 

On the contrary, both the American colonies and France were struggling against perceived injustices committed by their monarchs. The American Revolution was a revolt against "taxation without representation" which was considered tyranny. It is important to remember that England's purpose in colonizing America, starting with Virginia (which was originally the Virginia Company), was to make money. They collected revenue from successful crops and, later, taxes for goods imported to the colonies, such as tea.

France raged against the excesses of Marie Antoinette and Louis XVI. The lower-classes could no longer afford the price of bread while the queen naively dismissed the problem (she was very unfavorably represented by the press).

Key Enlightenment thinkers, such as Voltaire, Jean-Jacques Rousseau, Montesquieu, and John Locke provided ideas that offered a way forward. Rousseau had particularly influenced the French revolutionaries. He rejected Hobbes's cynical view on the state of nature, arguing instead that humans were once happy, good, and free in the state of nature and that society had ruined them. The problem, for Rousseau, lies in the constraints imposed by society, not by nature. 

He considers the original societies that were formed -- people building the first huts. This arrangement required the cohabitation of men and women (the origins of monogamy), as well as comparisons of one's own life and resources to someone else's (envy and competition). While Rousseau conceded that there were natural inequalities between people (e.g., some being more intelligent than others), it was society that made people feel inadequate as a result of these natural inequalities. Particularly fed up with the inequalities in French society, it is no wonder that the revolutionaries focused on this notion and used it to completely change their nation, with a focus on the values of liberty, equality, and fraternity.

After winning the Revolutionary War, the Americans focused on creating a new government, one that would not repeat the mistakes of its former keeper. English philosopher John Locke's ideas about about the importance of reason and tolerance as well as the making of laws devoted to the public good were key for the Founding Fathers. 

Voltaire was a key influence on the importance of free speech in a free society. Finally, the legal scholar Montesquieu's book, The Spirit of Laws, introduced the idea of a government separated into three powers, executive, legislative, and judicial, each of which would "check and balance" the other.