Originally? They're for carrying weapons to use to kill your enemies.
Eventually warships come along, including cruisers (which suck) and destroyers (which are used to haul fighters between systems). In the Terran-Vasudan war fighters (bombers, specifically) are used to drop bombs on your enemies, including their civilians, who you nuke en masse. Fighters are used to kill these bombers, or to kill other fighters.
They are also for guarding convoys full of profitable things against other fighters which could be used to steal them.
By the time of FS2 the balance of power is swinging a bit back towards the warship, but fighters are still the prime means of power projection, especially through FreeSpace's equivalent of an over-the-horizon attack, i.e. ops at least one jump away from your base.