Ok, picture this, dwarven hair is very tough and doesn’t fall out like it does with other races, with one notable exemption.
When a dwarf is dying their hair will shed slightly. Not enough that they’ll be bald by the time a pass into the halls of their forefathers, but enough to be noticeable. In fact, many dwarves find out their dying slowly by the hair in their brush.
So the whole point of this is that when Bilbo stays in Erebor, one of the dwarves find his brush, and the hair left behind in it.
Of course this sets all the dwarves into a frenzy and then some morning while a clueless Bilbo tries to understand what is upsetting his dwarves. He better find out soon,he doesn’t think he can survive another back breaking hug nor another bout of spontaneous crying when he walks into a room.