There were at least three ways in which cultural exchange took place between the Muslim world and western Europe.
First, there was trade. Muslims traded with the Italians in particular. The Muslims controlled trade with Asia and the Italians were the middlemen between the Muslims and the rest of Asia. Cultural exchange took place via that route.
Second, there were the Crusades. There was some amount of cultural exchange due to the fact that many Western Europeans came in contact with Muslims in the Holy Land. This exchange is generally exaggerated, however.
Finally, the most important point of cultural exchange was Spain. Muslims ruled parts of the Iberian Peninsula for hundreds of years. There was a great deal of contact between Europeans and Muslims during that time.