It doesn't matter if there's a literal reason for it or not, I'm talking about having the belt as a design choice.
Superman's red trunks break his blue suit into two sections, visually, which makes the whole thing much more appealing. The Flash has a yellow belt for the same purpose, and he's not carrying anything around either. The MOS suit didn't have anything to break up the design, and that proved distracting (see: every forum post since 2011 pointing and snickering at Henry Cavill's bulge).
He doesn't need to be wearing underwear, but he needs something near his waist to make the design easier on the eyes.