Dolphus Raymond is a white man married to a black woman. He, as expected is frowned upon by society, because not only did he marry a black woman, but they had children who were born mixed as expected.
However, Dolphus made a point of appearing to be the town drunkard in order to excuse himself in front of the town as to the oddity of his behavior in mixing himself with the blacks. He would carry his brown bag around swaying his walk and pretending to be drinking alcohol from the bottle inside the bag.
In reality, the bottle was soda and he demonstrated that to Dill, Jem, and Scout. He explained to them that if he uses drunkenness as an excuse for his behavior people won't inquire upon his choices anymore.
The meaning of this is that such was the extent of racial distance that existed in Maycomb. It also shows how, to the townsfolk you had to be either crazy, a "n-lover" (like they called Atticus) or a drunkard to even consider loving, helping, or even mingling with a person of the other race.