Your Barracks serve as a place for your followers to sleep.  However, the level of your garrison also affects the number of followers you can send on missions at a time.  While Blizzard has stated right now the numbers are in flux, it is looking like you can send out 5 followers at once for a level 1 Barracks, 7 followers for level 2, and then 12 followers for level 3 Barracks.

The interior has the actual barracks your followers will sleep in.