Essentially ...
The brown wire mod is an electrical modification that does two things:
Increases the gauge of the ground wire from the battery to the body in the front trunk; and
Runs a second, heavy gauge, brown wire from the positive side of the battery to the fuse box. That second wire parallels the primary one that's there, thus providing a more robust positive electrical path to the car's electrics.
Others will provide a more detailed description and there are numerous threads detailing the how and why of installing the brown wire mod.