Many reasons have been covered. The biggest point is that if you are showing stock horse breed shows, it is expected you will have one in the majority of classes (exceptions include halter, cattle classes, etc.).
They also, as mentioned, help quiet an active tail or encourage a horse to hold the tail down. Both things are desirable.
Another reason for a fake tail, especially a larger one, can be to help disguise a horse with not the best hock action. If you have a horse with a lovely front leg, but it doesn't move as nicely behind, a big fake tail can serve as a distraction and help cover up the hocks. Seeing as we are taking about horse shows, you want to do what you can to present the best possible image. Covering flaws is part of that, and a fake tail can help.