The water freezes - and so do they.

I also thought this... but nobody else seems to

If the water temperature gets below 0C, it freezes, nothing to do with it moving, (Rain freezes into hail)

well not really, if there are pipes to it then the pipes freeze and block the flow, if they are natural then the same thing can happen, they source the water is streaming from can freeze which i would think would cause it to not flow... maybe im wrong.

It doesn't freeze all at once. The process usually starts at the edges of the waterfall, where rime (a crust of ice) begins to form from water spray. The surface of the water above and below the fall may also freeze solid where it is calmer. Over time, the ice begins to advance inward, choking and slowing the flow of water. Once the water slows down enough, it can freeze solid, but often there is a tube of free-flowing water under all that ice.