Think of it in this way. It's the way I try to to make sense from what on the surface seems senseless, the stalemate.
By taking away all of your opponent's moves while he is not under attack, he cannot make a legal move. As a result, because he cannot move, you cannot proceed enroute to your victory either because your opponent has to move before you can.
In actuality it makes sense. Not only that, it behooves you to play the the to checkmate right or suffer the "embarrassment" letting your opponent get away if you're on the giving end, and it gives you a sense of relief and accomplishment (fist pump once or twice) if you manage to pull it off on the receiving end.
In Chinese Chess, a stalemate results in the stalemated player losing because naturally, if a king is cornered and has nowhere to go, the king loses. Why does European Chess go against intuition and produce a draw under that circumstance?