One good experiment that explains centripetal force is the spinning penny. A penny is put inside a balloon, and the balloon is filled with air. Then, the balloon is swirled around until the penny rolls around inside of the balloon. The penny continues to spin even after the swirling stops.
The reason behind the circular movement of the penny within the balloon is centripetal force. It is a force that is always center seeking, and keeps the penny's circular movement going even after the swirling has stopped. Other experiments that are used to explain centripetal force are the bucket swing and the ping pong challenge.