In the Atlanta Compromise speech, Washington tells the story that you mention. His point is to argue that both blacks and whites in the South need to find a way to live with one another instead of trying to look elsewhere.
He says that blacks should not try to leave the country or to go to the North. Instead, they should get along with the whites where they are. This is analogous to the ship's captain casting down his bucket--they should realize that they have what they need right there.
He also says that whites should make use of black people rather than looking to immigrants. He says that Southern whites know that blacks are hard-working and faithful and will not go out on strikes or cause any other problems. Therefore whites should realize that the labor they need is right there and "cast down their buckets" among the African Americans.