Breathing therapy for Cluster Headache involves inhaling high-flow, high-concentration oxygen (typically 12-15 L/min at 100% concentration) through a non-rebreather mask, which rapidly causes cerebral vasoconstriction and reduces regional blood flow in the hypothalamus. This therapeutic approach works by activating the trigeminal-autonomic reflex and rapidly alleviating acute cluster headache symptoms through its vasoconstrictive and neurological modulating effects.