ANCs are a product of Home Rule. Prior to Home Rule (via the Home Rule Act of 1973) Congress (the Federal government) ran the city. It wasn't until 1974 that DC residents were able to vote and have some real say in how the city ran. Before Home Rule the mayor and the city council were federally appointed. Neighborhood wise there were citizens (white) and civic (black) associations that appealed to Congress and city government officials for things like neighborhood improvements, traffic, crime and so forth. As far as I can tell civic and citizen association leaders were elected by the association's membership. These groups could only beg or appeal to bodies and officials whom they could neither vote for or vote out of office.
With Home Rule, neighborhoods got something new:
... the Advisory Neighborhood Commissions (ANCs), brought the city administration closer to ordinary voters than any other elective units. The city council created 36 ANCs and 376 smaller single member districts, each representing about two thousand residents. The ANCs were intended to serve as neighborhood mini councils that advised the council on local problems.
--City of Magnificent Intentions: A history of Washington, District of Columbia 2nd edition P.584