It doesn't actually burn the mark onto the person (at least I assume not, otherwise it would have the fire descriptor) so why does it do damage?
Magic!
YOU WANT THEM TO FEEL IT.
...I honestly think this is the reason. If you just wanted to leave a mark, use an arcane mark (it lasts longer- 1 month flat duration on creatures vs. 1day/lvl). So the purpose of the spell is likely meant to be more of an image thing, particularly since it is only an inquisitor spell by default.
In this case, it is the image of a branding iron. Used to hurt and shame someone. It allows the perfect number of characters for things like "LIAR" or "SINNER". You force the target to hide in shame or suffer excruciating pain to remove it (in a fashion that is ultimately futile, since it comes back if healed).
I would imagine whoever designed the spell in setting did it with the intent to make a spectacle out of its use, having the target scream in pain in front of a crowd.
Ok, that makes sense. Speaking of which, why doesn't arcane mark allow a save when writing it on someone? I can easily think of cases when you wouldn't want to be marked in such a way.
It doesn't actually burn the mark onto the person (at least I assume not, otherwise it would have the fire descriptor) so why does it do damage?
It doesn't burn, but it does cut:
Brand etches an indelible rune or mark of no more than 6 characters onto a creature, inflicting 1 point of damage.