I'm pretty sure muslims believe it was written by man, but directly told to by a prophet.
If you mean they believe a man wrote it down on paper then you're correct but if you mean written as in "a-man-made-it-up" then no, they don't believe that, they believe it was told to the Prophet -who couldn't read or write- by an angel named Gabriel and the Prophet told it to the people.
The main activity of the Arabs at the time -and still now in a lot of places around the Gulf- was poetry contests, second only to horse races, so they were high on memorizing and reciting insanely long stanzas of poetry and pronouncing it correctly by the guide of signs above the words that tells stuff about how the words should sound. (I exhausted my vocabulary and can't really tell you what the signs are for heh)
So those who entered the faith memorized it -in parts, the Quran was delivered over the course of many year but I don't recall how many- and a few years after the prophet's passing; the Khalifa ordered it to be written on paper for future generations, it had 184 men confirming it's accuracy at the time before making it available to people.
Khalifa is the direct translation of successor, Western people refuse to translate it into English for some reason, just like they don't translate Allah into God which is also the direct translation.
The "message" was finished just before the Prophet died and he told the people that he "Delivered the message", Muslims call the Prophet "The Messenger" in Arabic "Al-Rasul".