Initially, God breathed a soul into Adam through his nostrils (Gen 2.7). So we definitely know that breathing (נשימה) is linked somehow to the soul (נשמה).
We require breathing to survive and without it, we die pretty much immediately; the capital punishment Heneck is based on this phenomenon.
But, how do the sages explain the need to breathe, both physiologically and spiritually?
Physiologically: what does it do? Air is not mentioned in the Torah, and IIRC was only "discovered" by the Greeks (see Air). So, unlike the physical food that we intake and digest, and that fuels our bodies, it is not clear what air is needed for.
Spiritually: if breathing relates to the soul, maybe to feed our souls we need to breathe in additional souls or parts of them and to exhale some parts of ours, and without that, the soul "starves" to death (just a suggestion).