It's seen quite clearly here, with the front part of the harness removed. (This is a training harness). The breeching is, in fact, part of the braking system. How does that work?
When the horse stops, the cart or carriage doesn't - right away, anyway. The shafts then pull the breeching forwards, into the animal's butt. The horse then applies pressure to the breeching, which helps stop the cart. Sometimes with very light show carriages you won't see breeching used, as the cart is so light the horse can stop it from the shafts and saddle alone.
Breeching is almost never used or needed on saddle horses (if a horse is prone to the saddle slipping forward, a crupper is used - I'll talk about those tomorrow). This isn't an exception - these aren't horses! Mules have flatter withers and it's not common to use breeching, as shown here, on saddle mules in rough terrain.