How did Islam lead to European Exploration?
As severusm00 mentioned, Crusaders brought back all kinds of exotic goods with them from the Holy Land. From the 11th to 14th Century, Europeans developed a taste for foreign spices, textiles, and precious metals. When the Crusades waned and eventually came to an end, demand for these goods in Europe didn't cease. This laid the foundation for extensive trade across Europe and Asia. From the farthest reaches of China, Southeast Asia, and the Indian subcontinent, goods were traded towards the Middle East. There, Ottoman traders and merchants—who practiced Islam—would bring the goods for trade in Italy. Finally, from Italy, these goods were traded throughout Europe.
The Ottoman Empire was massive; it stretched from the Balkans and North Africa, well into the Arabian Peninsula. Once goods made their way to one of the many trading points of the Ottoman territory, these merchants knew that they could fetch a good price by trading them to Europeans. Eventually, Europeans realized that with new advances in navigational technology and shipbuilding, they could find their own way to exotic lands. Much of European exploration had the goal of "cutting out the middle man" by acquiring trade goods directly from the source at a better price than if they had come by way of Ottoman traders.
As you can see, Islam was influential to European exploration in two ways. First, the Crusades were a series of military campaigns against Islam in the Holy Land. Islam was the "reason" people went to the Eastern Mediterranean and consequentially brought back exotic goods. Second, Islam was tangentially related through Ottoman, Muslim traders who sustained the demand for exotic goods in Europe.
When the Crusaders returned from the Middle East, they brought silks, spices, and new educational ideas with them. These finer goods started a frenzy among Europeans. However, Europe wasn't able to produce these goods. They looked to other undiscovered areas (specifically Asia) for these goods.