Like an airplane wing or the dimples on a golf ball, a frisbee is designed to create a reaction force to the surrounding air. You throw the frisbee.
Gravity pulls the frisbee down.
But the frisbee's round flat shape, its spinning motion and the angle at which it's thrown all act together to force the air beneath the disc down. The frisbee couldn't work as an airfoil. It wouldn't fly any farther or more smoothly than, say, a moon rock of the same weight. But, in fact, because of the lack of air and the moon's weaker gravity, a frisbee would go perhaps ten times farther on the moon than a frisbee thrown with the same force here on Earth. And, by the way, Kim, airplanes couldn't fly on the moon.