it's from baseball - the plate is the place where the batter stands to bat, so when he steps up to it he is preparing to do his job, show what he's made of etc. American, like many common phrases these days.
I was told that 'the whole nine yards' refers to the length of a machine gun belt in the wing of a WW2 fighter plane.
Thus when you had used all your ammunition you had 'gone the whole nine yards'.