The underside of something is its bottom surface — the face that points at the ground, or the face you cannot see from where you normally stand. The underside of a leaf, the underside of a car, the underside of a plate, the underside of a bridge. It takes 'the' almost every time, and 'of' follows it naturally, because an underside always belongs to something. Notice how it differs from 'bottom': the bottom of a box is the part it rests on, which you might think of from inside or outside, but underside is specifically the outer face seen from below — the view an insect on the floor would have. That is why so many uses involve looking, checking, hiding: you turn a leaf over to check the underside for insects, mechanics look at the underside of a car for rust, and somebody tapes an envelope to the underside of a drawer. English also uses it figuratively for the hidden, uglier part of something otherwise pleasant: "the underside of the city's economic miracle" means the poverty and hardship underneath the success story. That figurative use is a bit literary but common in serious writing.