Rate of Fire is the frequency at which a specific weapon can fire its projectiles, or magic attack for wands. This correlates to your ranged attack speed. Ammo pouches and quivers grant a speed increase for storing bullets or arrows in a ammo bag instead of a normal bag. All ammo bags increase your ranged attack speed (10%-15% depending on the item).
Rate of fire directly affects the speed of attack for weapons for which the character has the autoshot ability (mages, priests, and warlocks with wands, and hunters with bows, crossbows, or guns). For shots or throws which must be manually initiated, the manual triggering process tends to overshadow the rate of fire; still the manual shot is available sooner.