'The flat of' means the flat part of something, such as a hand.
If you beat/strike someone repeatedly with the flat of your hand, or other object, they will become bruised. Bruises are black and blue in colour. Get it?
Look HERE for an explanation of the terms used with swords (or at least one particular type of sword). As you can see from the second picture, the flat side of the blade is actually called "the flat", so it is a noun at the end of your sentence.