Why did the American and French revolutions happen?
Even though there were major differences both started as tax revolts. The American colonists didn't think that the King should be able to impose new taxes without their consent ("no taxation without representation). The colonists had a long history of self-government at that point, and their original goal was not independence from Britain but a say in matters that affected them directly. After the fighting started and positions hardened they finally decided to declare independence.
France was rather different. Directly because of the 7 years war (known here as the French and Indian War) and French assistance in the Revolution, and because of their inefficient tax system, (the clergy and nobility were exempt from many taxes that affected the peasantry and middle classes) France was, to all intents and purposes, bankrupt. King Louis the XVI called the Estates General into session for the first time in over 150 years to vote new taxes. There were 3 estates, the 1st estate, the clergy, the 2nd estate, the nobility, and the 3rd estate, the commons (everybody else). Each estate had a 1/3 say in what was to be decided. The clergy was dominated by the bishops and archbishops of the Catholic Church, and had noble status. Since the first 2 estates interests pretty much aligned, they could control the outcome, even though they represented less than 10% of the population of France.
Obviously the representatives of the 3rd estate did not like the arrangement. They withdrew from the proceedings. When the King, Louis XVI locked their hall and posted armed guards outside of it in an attempt to stop them from proceeding they withdrew to a nearby indoor tennis court. There they swore an oath not to disband until they had drawn up a constitution for France. This is generally considered the beginning of the French Revolution by most historians, and is known as the Tennis Court Oath.
Contrary to popular views of the French Revolution it wasn't a simple case of exploited peasants rising up against oppressive aristocrats. The revolutionaries were mostly middle class intellectuals, and a fair number of aristocrats and clergy (particularly lower clergy, such as parish priests) joined them in the tennis court. Also, as more radical revolutionaries displaced some of the more moderate ones they set out not to just give France a constitution but to radically remake French society. French society was to be de-Christianized, and priests were required to swear an oath putting allegiance to France above allegiance to the pope. Ironically this helped set off a horrible rebellion in the Vendee region of France by devout peasants, who feared that if they received the sacraments from priests who had sworn the oath that they would be endangering their souls. The rebellions were put down with great brutality, with revolutionary armies slaughtering whole villages, and atrocities being committed by both sides. Estimates of the death toll range from 100,000 to 250,000. The actual revolt started with a conscription act.
Unlike the American Revolution the French Revolution ended rather badly, with many of the more moderate revolutionaries being executed by the more radical ones, and then the radical ones, such as Robespierre and Louis de Saint-Just, being guillotined in turn. Eventually the Directorate, the successor to the Committee of Public Safety, was overthrown by Napoleon Bonaparte. This started a period of almost constant warfare in Europe with a level of death and destruction not seen since the 30 years war in the 17th century.
Sources: Wars of the Vendee | French history
The nutshell answer is this: "Each nation wasn't happy with their rulers, so they kicked them out."
In the case of the Americans, the British discovered that it's hard to fight America and France at the same exact time.
In the case of the French, when the queen believes that the bread stores are just out of stock and asks "Why don't you just let them eat cake?" without realizing that there's widespread famine; the queen loses her head.
Each of the two revolutions happened for different reasons, but there's a little bit of truth value to the idea that the French Revolution happened because of the American Revolution.
Firstly, there's the possibility of "He did it, so I can do it too!" in which the French saw the Americans' idea, and decided to emulate that. What a magnificent compliment.
Secondly, after King Louis VIII spent a huge fortune supporting the American troops, his people were furious at the cost of war with England. They spent their money to support us, and without that money to support themselves, the citizens threw a tantrum and killed the king, the queen, and a long list of other people afterwards.
This is not a subject that can be covered in a Quora post. It would take several pages to outline the various factors involved in both of these revolutions. I would suggest that you Google each of these. There will be reams and reams of information.