They are clipped to the horse's headcollar or halter to insure the horse is secure whilst travelling, or when being groomed or tacked up.
When securing a horse with a lead rope for any purpose you must always use a quick release knot to allow the horse to be untied quickly in any emergency.
Tying a horse to a fixed object such as a gate can be dangerous because the horse could injure itself if it were to pull back suddenly.
Instead tie the rope to the fixed object with a piece of bailing twine. So that if the horse pulls back the twine will break and the horse will not hurt itself.
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