Aluminum dissolves in the soda-water solution, and silver reprecipitates on the articles. In other words, the silver sulphide (black) gets reduced to the metal by the aluminum, which is more electropositive. In order to conduct, the solution has to be a bit alkaline (sodium bicarbonate or carbonate). Your metal plate is possibly made of zinc. Aluminum foil works just as well and is cheap, but in either case some silver will be lost, so don't overdo it.