A tongue tie is a strap that's used to restrain the tongue and hold it in the lower part of the mouth when a horse is being ridden or worked. They are designed to prevent the horse from getting its tongue over the bit or otherwise interfering with the bit action. (It's also often said they help keep the airway clear, but I'm not sure how as horses do not breathe through their mouths).
Tongue ties are commonly used in racing (when control is paramount). They're illegal in many other sports and often frowned upon as they can cause severe injury to the horse's tongue if not fitted correctly.