What is it about heat that gets rid of the wrinkles?
Heat can also make wrinkles, e.g. a curling iron. I think the heat allows the fibers to set in whatever configuration they are in at the time, and since the iron and board are both flat, the fabric sets flat too.
but why?
From How stuff works
But you can iron out wrinkles without water or steam.
Not very well.
sure you can