Cliff notes for those without the time or who would like to know what's in it before giving it a listen:
Islam stands upon 3 pillars - the Quran, Muhammad, and Mecca. If any of those 3 falls, all of them fall due to the intertwining of the 3 into the history of Islam. As it so happens, all 3 of them lack supporting evidence to show they existed at the time when Muhammad supposedly lived.
Starting with Mecca, the book "Meccan trade and the rise of Islam" details the studies of Dr Patricia Crone - unique among scholars for her ability to read a number of ancient languages from the Middle East around the 6th century - and shows that the city now known as Mecca did not exist until at least 70ish years after the supposed death of Muhammad. Crone goes through every available manuscript looking for mentions of the city of Mecca and finds nothing. Instead, she manages to find mentions of every settlement large or small surrounding Mecca that are contemporaneous to the life of Muhammad. Given that Mecca exists in a desert with no easy access to water, this is a serious blow to the credibility that this city existed from 570 to 632.
This alone is enough to discredit Islam, which claims the perfect preservation of the Quran across time by God. But it goes on. The book has serious problems with origin as admitted to by Dr. Yasir Qadhi. Far from being perfectly preserved across time by God, the Quran actually has no less than 30 surviving distinct Arabic versions. The total number of textual differences between these versions number in the hundreds of thousands. The original manuscripts, of which 6 are held as the oldest, do not begin to show up until over a century after Muhammad's death - a curious fact considering that the Caliphates of the day were the wealthiest kingdoms in the world and had access to plenty of animal skins to record the Quran. Instead, what we find is strong evidence of conflicting versions early on in Islam's history followed by attempts to censor and destroy these variant manuscripts.
Lastly, the supposed prophet himself, Muhammad. As it turns out, this name is not recorded in written history until much later than when Muhammad supposedly lived and died. This is due to the early written Arabic language lacking vowels - it possessed only consonants. Thus the closest to Muhammad that can be found at the time is MHMD, but this presents a problem. In the 7th century, St John of Damascus writes in Greek (a complete language with vowels) about a religion in Arabia that he describes as the Heresy of the Ismaelites, who follow someone he calls Mahmed. However, this name is more akin to a title or an honorific, meaning Praised or Anointed. It is particularly interesting that this title is actually seen centuries earlier in the Old Testament of all places, and thus opens up the strong possibility that the man people call Muhammad is something like King Arthur in that he is a collection of stories about men doing noteworthy things all attributed to one man.
Dr Jay Smith explains this all better than I can. Give it a watch, I think it's very convincing.
