You need a familiar to be initiated into magic
You can't become a mage without the help of a familiar. Whatever special quality a person has that allows them to perform magic can only be initially awakened by that person's familiar, so it's the first (mandatory) step towards all other magics.
Familiars store magical energy
There's a limit to how much mana a given person can accumulate and expend at a given time (although those limits might be different for different people). Familiars can also accumulate mana, and mages who use familiars can expend their familiar's energies as well to make their spells more powerful.
A familiar is like training wheels
Some mages are skilled enough to do magic without a familiar. Others need a little bit of help to properly focus their magic. A familiar doesn't do anything that you couldn't have done on your own, but having a familiar to metaphorically hold your hand while you're casting spells you makes it a that much easier.
A familiar is a "free" servant
Familiars can do other things besides just helping you do magic. They can also run errands for you or perhaps do simple chores. Why are familiars willing to do these things? Maybe they appreciate having a physical form and are happy to show their gratitude this way. Maybe they secretly siphon away a mage's soul bit by bit. Maybe they just think of mages as pets that need to be cared for.
It may be difficult to relate to people who aren't mages. And mages might view each other more as rivals than as friends. If that's the case, it might be nice to be able to summon someone (or something) who understands magic well enough to relate to you, but who isn't going to try to steal your ideas or usurp your power.