If you move it, you're really moving some of the atoms at one end. They 'bump' into nearby atoms, which 'bump' into more atoms, and so on, and eventually the atoms at the far end get 'bumped', and the far end moves.
Sound waves travel in the same way. So if you move one end, that movement propagates at the speed of sound (in whatever material the lever is made of).
Hmm, I'd take your word over mine but this doesn't fee quite right to me. What I'm thinking is, say you reverse the action so that instead of bumping one end of the see-saw, you release your grip on it having been holding it down. So in normal circumstances the other end would fall while the near end rises (of course now an external gravitational field has entered the thought experiment). But if movement is still due to the bumping of atoms , how does it work, where does it start? and if it's something else, then what, and is it still not instantaneous?
Sound waves travel in the same way. So if you move one end, that movement propagates at the speed of sound (in whatever material the lever is made of).