The black/brown color is because silver chromate is in the cells.
As you say, the precise mechanism is unknown but this is more about why exactly silver preferentially precipitates within neurons by this procedure, and not other cell types, and also importantly why it does so _so sparingly_ \- the key to the Golgi method is that only a minority of all neurons actually get labeled, and get labeled very strongly. If all neurons and processes would be labeled, you would not be able to see any of the details in a single cell.
Pasternak, J. F., & Woolsey, T. A. (1975). On the “selectivity” of the Golgi‐Cox method. Journal of Comparative Neurology, 160(3), 307-312.
Scheibel, M. E., & Scheibel, A. B. (1978). The methods of Golgi. Neuroanatomical research techniques, 89-114.
Špaček, J. (1992). Dynamics of Golgi impregnation in neurons. Microscopy research and technique, 23(4), 264-274.