At the request of the Archangel Michael (the archangel who mythically fought evil on dramatic hill-tops) and had its origins in 7th century Ireland), Bishop of Avranches built a church on the island in 709. In 966, at the request of the Duke of Normandy, a community of Benedictine Monks settled on the rock. As the centuries past, the humble church was developed into what is known today as the Romanesque abbey church.
The Abbey was built in the 11th century on a set of crypts at the tip of the rock and were the first monastic buildings to be built up against its north wall. The monastery buildings were extended in the 12th century to the west and south of the rocky island.
Mont Saint Michel is a World Heritage Site that attracts millions of visitors every year, for more on its history and when you can visit, please see http://www.ot-montsaintmichel.com/en/histoire.htm
Incidently, Mont Saint Michel has a story that runs pre 709 and its connection with St Michael’s Mount in Cornwall is certainly of interest. Christopher Long, a British journalist, takes up the story in more detail here http://www.christopherlong.co.uk/oth/msm-smm.html