The ride starts out with the platform flat on the ground. When you and the riders are in your places the platform begins to rotate faster and faster until it reaches a certain speed. At this speed the floor drops, but you don't drop with it, instead you seem stuck to the wall. A similar ride known as Round-Up or the Tilt-A-Whirl where the platform is tilted. However, you don't fly out. How come? It's all because of a force known as centripetal force.
Due to Newton's first law of motion inertia tries to keep you moving in a straight line, which would be away from the ride. (NOTE: The path that inertia wants you to move in is actually tangential to the circular path of the ride. We will talk about this later.) Fortunately, you don't fly out or drop with the floor because the wall prevents you from doing so. One way to look at this is that the you are being flung out but the centripetal force, exerted by the wall keeps you in the ride. Remember that the centripetal force is directed to the center. This is the reason why you seem to be stuck on the wall.
The ride in figure 1 also demonstrates centripetal forces which pins you down to the seat.
Here's a second example, swing rides.