The beach-based success is easy to understand; we have beautiful weather and beautiful sand, so tourists come visit all year long. The windiness of the island also attracts people interested in water sports like surfing, windsurfing and sailing. Each year the island even hosts the World Windsurfing Championships on its southern edge.
The cheese makes less sense until you think about goats. Yes, goats. While most of the cheese you eat likely comes from cows, you can make cheese from any type of milk. Here, where there isn't enough grass to keep cows happy, goats provide the majority of the milk to make cheese. Goats are easier to keep, because they need less space and can handle the more difficult climate presented by the desert. Not to mention that the rocks and trails are a better habitat for them.
So there you have it! Fuerteventura is a desert in the middle of the ocean, hit by hot winds, that people have figured out is the perfect place for windsurfing, sunbathing and cheese-eating.