What the well-dressed horse is wearing...
This is a "sleezy".
It isn't snapped under his chin because it's supposed to go under his halter. K was just trying it on him, so she left the halter on. She wasn't sure if he'd cooperate, but he was patient and willing. You roll it up like a t-shirt and put it over the horse's head.
A sleezy keeps the horse's mane in place, untangled, and clean, before a horse show.