What inflammation is is the engorgement of tissue with blood in response to that issue releasing a signaling chemical (histamine) that dilates blood vessels. That is why inflamed tissue is redder than usual. Theory is that the body has this response to rush additional white blood cells to the site of a lesion. (An allergic reaction is one where the tissue releases histamines in confused response to something innocuous.)
Tissue that is full of blood is thicker, heavier, and stiffer. Since vocal sound production is actuated by vibrating tissue (the vocal cords), when that tissue is inflamed, the vibrating apparatus is thicker, heavier, and stiffer. In the same way a thicker, heavier, stiffer guitar string sounds lower than a thinner, lighter, more flexible one, vocal cords which are afflicted with inflammation will sound lower than they otherwise do.
P.S. Singing on inflamed vocal cords will make the inflammation worse, by irritating them further.