The French Revolution was a truly multi-causal event (or series of events more accurately), combining economic failure, social unrest, political battles and idealogical change.
Based on the pre-Enlightended thought of Hume, Locke and Descartes, the Enligtenment movement embodied the beginning of a shift away from the popular ideas of absolute sovereignty and the rule of religion in society.
The rationalism of Descartes and Kant paved the way for a new philosophical model of the world, one now based around the individual as opposed to the collective. The idea of individual rights began to emerge and Rousseau articulated the idea of a social contract between monarch and people. Voltaire launched scathing satirical criticisms against both the monarchy and the Church, weakening religious sentiment in France. This was furthered by the increasing relativism of 18C Europe, with discoveries of distant societies and Diderot's expansive Encyclopedie.
Whilst it began as an intellectual movement, the ideas soon were proliferated and the basic sentiments, if not the philosophical reasonings, spread through society via pamphlets, images and discussion in coffee shops. Enlightened thought became the new vogue.
What is important to recognise here is not that the Enlightenment brought down the French Monarchy, but the development of individualistic and scientific thought questioned and challenged the religious absolutism of the French Monarchy. The undermining of the monarchy's rationale for rule was a key aspect in the eventual Revolution.