I always thought that the pieces were meant to represent entire legions actually and how they were used almost the same way as the armies of the medieval times were used.

The pawns being the infantry made out of poor levies just thrown at the ennemy to crush them under their number.

The knights being the cavalry parts, specialized in fast and powerful attacks.

The rook meant to be the artillery and siege equipment, firing them in a straight line to punch trough the defenses by brute force.

Bishops (which go by names like "Fou" which means "Fool" in french or "Läufer" in German which means "runner") maybe meant to represent maybe archers or standard bearers? (since they stand close to the main officers)

The Queen being more of the King's right hand, doing eveything in his place.

And the King being a king, basically useless but you have to protect him because he's a douche and everyone wants to kill him.

On that note, here, have some nostalgia. It's on the house.