He was called Ayasya Angirasa, for he is the sap (rasa) of the limbs (anga). Verily, breath is the sap of the limbs. Yes, breath is the sap of the limbs. Therefore from whatever limb breath goes away, that limb withers, for breath verily is the sap of the limbs.

In plain words: The breath is called Ayasya Angirasa because it is the life-juice of the limbs. Whatever part of the body the breath leaves, that part dries up and withers.