A Frisbee flies because its design creates a low-pressure area on top of the object and a high-pressure area on the bottom, giving the Frisbee lift. The thrust is given by the arm of the person throwing it, while the spin imparts stability.
Frisbees fly because of the shape of the object and the kinetic energy imparted as it is thrown. The rim of the Frisbee allows air to move over the top of the object faster than the air moving below it. The faster moving air on top of the Frisbee has a lower pressure than the slower moving air across the bottom, causing the Frisbee to fly. The spin of the Frisbee as it flies gives the object stability through angular momentum, keeping it level and maintaining a smooth and steady flow of air around the Frisbee. The stability imparted by the spin also ensures that the Frisbee sustains the proper angle of attack, which is the ideal angle at which the Frisbee much fly to ensure the most lift.
The principle of the Frisbee's design is easily tested by turning it upside-down and then throwing it as normal. The Frisbee can still fly when thrown inverted, somewhat, but is much more unstable and the amount of lift generated is significantly less.