Humming induces vibrations in your paranasal sinuses (the air filled spaces which surround your nasal cavity). This oscillation causes an exchange of air between the sinuses and nasal cavity.

The paranasal sinuses produce increased Nitric Oxide which is circulated to the nasal cavity (link). Inhaling in through your nose following humming will distribute the increased Nitric Oxide into your lungs, where it may inhibit pathogens, viruses and bacteria. Here is a useful technique you can try (link – begins after 10 minutes).

https://www.youtube.com/watch?v=LpK_sbxEcKU