It seems to be because the sensation induced in the mouth is literally that of heat. I remember Adam Ragusea on his excellent YouTube channel pointing out that capsaicin makes the mouth much more sensitive to temperature. So funnily enough, the sensation of something being too hot temperature-wise, and something being spicy is pretty much the same thing. It's also why cold water seems to help for a bit, but does nothing to eliminate the effects of the capsaicin as opposed to something like milk.