The major idea of the Enlightenment was that the world and human life should be governed by rational rules. The thinkers of the Enlightenment believed that things could be understood through reason and not through reliance on supernatural factors. They did not believe in simply accepting the word of authority, wanting instead to see empirical proof of things before they accepted them. These ideas were not compatible with the Catholic Church of the time.
In a sense, the Enlightenment was likely to conflict with any religion. God cannot be proven to exist and anything that God does to intervene in the world (like through causing miracles) would go against the rational laws of the universe. Therefore, the Enlightenment and the Church were sure to come in conflict over the whole idea of God and what God's role in the universe is.
The Enlightenment was also opposed to the Church's insistence on hierarchy and authority. There was no rational reason, for example, to believe that the Pope has been chosen by God and given infallibility. On this front, too, the Enlightenment and the Catholic Church came into conflict.
The above answer misconstrues the opinion of Enlightenment thinkers. They were not opposed to religion per se, in fact, Voltaire, the famous Enlightenment thinker once commented:
If God did not exist, it would be necessary to invent him. I want my attorney, my tailor, my servants, even my wife to believe in God, and I think that I shall be robbed and cuckolded less often.
Enlightenment thinkers had no problem with the idea of God; however they tended to believe that God was an impersonal force who set the universe in motion and did not interfere with its operation. This belief was called Deism. Thomas Jefferson was also a Deist: note his use of the term "Nature and Nature's God" in the Declaration of Independence.
The questioner correctly asks about Enlightenment opposition to the Catholic Church, not Christianity or religion--a point which the first answer overlooks. Their problem with the Church was the many abuses which had been committed in the name of religion. Voltaire often commented in his writings on the Church, Ecrasez l'infame! ("Crush the horrible thing!") Of priests, he said, "they sing, they eat, they digest." He also made a satirical comment about Chinese people in an obvious reference to the Catholic Church:
They have an admirable religion free from superstition and the rage to prosecute.
Only one Enlightenment thinker, Denis Diderot, was a professed Athiest. All others considered the idea of God quite rational; they simply believed that God had created a rational universe.