It is considered proper to wear an undershirt under a button-up shirt here for several reasons.
1. If you sweat, the sweat gets on the undershirt not the good shirt.
2. When you do sweat, clothes tend to cling to the body. With an undershirt the undershirt clings but the actual shirt doesn't cling (unless you are sweating like crazy).
3. Some people I have heard claim it adds body to the shirt.
As for an undershirt being too hot and uncomfortable, actually I find the opposite, I am far more comfortable with an undershirt than without, including temperature-wise.
As for me, I never got it/like it either as I mentioned above. BUT, I can see if as a fashion... showing off a little undershirt (of whatever style) with a shirt unbuttoned a little. The only really overwhelming reason I can come up with, though, is so that your nipples don't show through a thin or white outter shirt.