Then they (the Devas) said: 'Where was he then who thus stuck to us?' It was (the breath) within the mouth (asye 'ntar), and therefore called Ayasya; he was the sap (rasa) of the limbs (anga), and therefore called Angirasa.

In plain words: The gods asked who had stood by them so firmly. It was the breath inside the mouth, so it is called Ayasya. It is also the life-juice of the body's limbs, so it is called Angirasa.