The French Revolution was caused by a number of factors, some of which served to exacerbate others. Some of these factors, like a dying class system, were common to most of the European revolutions. Philosophical shifts, national debt, and even the environment played a part in this rapid social upheaval, too.
Let's begin with the feudal system. For centuries, French society had been organized into three Orders or Estates: the laborers or peasants, the nobility, and the clergy. Under this system, the king divided up all the land in his kingdom among the nobility in return for military service. These nobles might then break it up among lesser nobles if they had the opportunity. Again, this obligated the lesser nobles to military service. Finally, the land was worked by peasants who had little say in their work or life situation. The clergy's function in this society was essentially to reinforce this system of social stratification by maintaining that it was how God wanted things.
By the time of the Revolution, the feudal system was weakened in many areas, and confusing in others. A shift in philosophies of the time (called the Enlightenment) and increasing literacy rates resulted in a peasant class who were acutely aware of their exploitation. Several years of crop failure due to weather had resulted in the peasant class being unfairly punished for something they had no control over. The food that peasants produced "belonged" to the king and nobility. When crop failure resulted in food shortages, the nobility were just fine, but a rise in prices put poor laborers at an unfair advantage. They couldn't afford the foods they were raising and producing! A number of bread riots broke out in protest of the high price of bread, a staple food many relied on.
Enlightenment philosophy emphasized equality and independent thought. It played a major part in the American Revolution, and in turn influenced the French Revolution. France was doubly impacted by the American Revolution-- they had fought alongside the United States against the British, and this military involvement had driven the government into extreme debt. Most of French society was unhappy with this fact and felt that it showed irresponsibility on the part of the monarchy.
So, what happens when a starving and exploited peasant class no longer feel obligated to participate in a weakened system of societal organization? Revolution!