Rooms are separate areas of your pawn shop that add bonuses to your shop and space you can use.
The mini-map will take you to any rooms you've unlocked, and allows you to navigate from room to room with ease. The mini-map will expand as you purchase more rooms.
The back room serves as a storage area for special or extra items, while holding certain improvements that can only go in the back room.
Expansions make your shop bigger, increasing the amount of floor and wall space you have.