It's not an assumption. If the commentator sees the horse is being restrained by the jockey holding a tight rein then the horse is "on the bridle". When the jockey gives the horse free rein to run as fast as he can then the horse is "off the bridle".
You can quite clearly see this for yourself, you don't need to be a racing expert.
This is the definition given by Newmarket.
ON/OFF THE BRIDLE: A tired horse reduces his effort and is said to be `behind or off the bridle". A horse with `plenty of running' still has energy and is still in contact with the jockey via the bit and the reins and is said to be still `on the bridle'