A wiglet reduces the loss of lift at the wingtip producing a more efficient wing. Better wing equals better fuel economy.
The winglet's function is to break the vortex that forms at the tips of the wings. A vortex creates substantial extra drag. If you've heard of ground effect, where an airplane has less drag close to the ground, this is because the ground breaks the vortex. Winglets have this same effect, though require high structural strength at the weakest part of the wing.
Thanks for the info bandit!