Muslims believe it for several reasons: the Quran's unmatched style, its internal consistency over 23 years of revelation, its accurate teachings about God and the human being, the changes it made in those who followed it, and its own challenge to anyone who doubts it to bring something like it.
The fuller answer
The Quran invites people to reflect on it and says that if it were from other than God they would find in it many contradictions. It makes a challenge to produce a chapter like it, and Muslims say that the challenge has been open for 1,400 years. Its Arabic is considered a miracle of eloquence even by non-Muslim Arabic scholars.
A Muslim's conviction is not only intellectual. Many describe a personal experience of reading it and finding in it answers to their own questions. Muslims welcome people to read it, in translation, with an open mind and ask questions of it.
And if you are in doubt about what We have sent down upon Our Servant [Muhammad], then produce a surah the like thereof and call upon your witnesses other than Allah, if you should be truthful.
Say, "If mankind and the jinn gathered in order to produce the like of this Qur'an, they could not produce the like of it, even if they were to each other assistants."
This answer follows the mainstream Sunni understanding and is general learning, not a fatwa. Where Muslims differ, the answer says so. Have a question of your own? Write to us.