The adoption of the crescent-and-star symbol for Islam occurred quite late in Islamic history, probably in the nineteenth century AD.

Most likely, this was due to European influence; because the cross symbolized Christianity, Europeans felt there should be a symbol for Islam. So, this symbol was adopted and became popularized.

So there is no inherent connection between the crescent moon-and-star symbol and Islam. However, crescent moons and stars have been used historically in Islamic art, on coins, and on banners as a cultural symbol. (They also appear worldwide on art in general.) 

Of course, the crescent moon symbolises the lunar calendar, especially marking the beginning of the month of Ramadan and Eid al-Fitr, so possibly this is one reason it was accepted.