"While there is no mention of FGM in the Quran, a Hadith (saying about the life of the prophet) recounts a debate between Muhammed and Um Habibah (or Um ‘Atiyyah). This woman, known as an exciser of female slaves, was one of a group of women who had immigrated with Muhammed. Having seen her, Muhammad asked her if she kept practicing her profession. She answered affirmatively, adding: “unless it is forbidden, and you order me to stop doing it.” Muhammed replied: “Yes, it is allowed. Come closer so I can teach you: if you cut, do not overdo it, because it brings more radiance to the face, and it is more pleasant for the husband.” - WordPress
This is, indeed, the passage Muslim clerics cite when they say Islam demands this of women.