Namely, giving us a route in. And a comparison. And someone to sympathesize with -- they often are much nicer people than the genius is. Which means they also give the main character a reason to do things, by providing moral pressure.
One audience member thought they anchored the character. Watson was more tied to nineteenth century England, and the companions to modern England. Whereupon I pointed out that the Doctor used to have a lot more variety in his companions -- they didn't have to be from Earth, and even those who were, didn't have to be contemporary to the filming. I observed that for all the cheap FX, the shoddy scripts, the lack of casting budgets, they were clearly having a lot more fun then, and panelists, many of whom seemed to agree about the relative merits, talked about how the more varied companions reflected a much wider world.
The Doctor's companions also could have character arcs, through their interactions with him and each other, and when they left, why you could have a new one that would be fresh because the character was new.
Plus, of course, the info-dump purpose. One panelist thought it was their main purpose. I pointed out that the writers really hated the episode "The Deadly Assassin" because the Doctor had no companion and no one to talk to.
Then, there are others. Many riffs off it. However, some audience members seemed to put forth any pairs of people - Buffy and Willow, where Buffy is not really a genius -- or any genius, such as in Girl Genius, where Agatha doesn't have a companion. (Moloch shows shades of it, but he's only intermittantly with her, and came in late)
Some discussion about the Scooby Gang approach, where there are several. The difference between Star Trek, where Kirk and Spock had the relationship, and The Next Generation, where there were many characters. The Doctor's occasional ensembles let the characters react to each other, as well as to the primary character.