It used to cover the payload carried by the stage below it. It makes the rocket more aerodynamic, thereby allowing it to travel at a higher velocity as it races to orbit. This also allows the rocket to be more fuel efficient. It also protects the rocket from air resistance, which causes overheating and exploding.
As of game version 1.8.0, only the Explosive Module and the end of a Resupply Module Dockable are small enough to fit under this tiny fairing, thus it is primarily used to cover rocket stages or small station parts directly to prevent overheating during launch.
Note : Fairings cannot be released during a launch until you have left the atmosphere and reached orbit.