How did Islam become such a powerful religion and cultural force in the middle ages?

You can start with Wikipedia:
http://en.wikipedia.org/wiki/History_of_Islam

My impression is that Islam had a lot going for it.  Although there
were civil wars, the religion with its belief that it was/is the only
true faith inspired political expansion and conversion.  The
acceptance that the civil and religious roles were interrelated
(Califat) tended to avoid the types of strife between the Church and
temporal leaders that occurred within the Christian realm from Byzanz
to Portugal.  Furthermore, at the time, the Muslims had the better
military prowess and also better scientific and philosophical
knowledge.

But maybe others have different opionions.

You should watch the 3 part series on PBS titled Islam: Empire of
Faith. That should answer your questions. As I am not from the US, i
have no idea if they are airing it again BUT there are resources on
the internet where you can acquire the broadcast.