1. They look cool
2. They provide natural ambient light that helps reduce stress on animals at night. They're not used to being in complete darkness at night in the wild.
3. Preditors are more likely to sneak up on an unsuspecting fish in a completely dark tank. With moonlights the fish can be kept more aware of preditors (if you have fish that are not completely compatible with each other).
In my tank the fish find nice hiding spots at night with the moonlights on. They can see where to hide and they go there. Before I got my moonlights, they would just "freeze" in the tank. When the lights would come on in the morning, they would act completely disoriented for awhile until they calmed down. My green chromis would flutter its fins and looked nervous when the lights first came on. It doesn't now that I have moonlights to keep it company at night.