Question

In his speech Booker T. Washington urges blacks and whites alike to cast down your bucket where you are. What does this mean?

Answer 1

To cast down your bucket where you are means that you don't need to travel far in order to find what you are looking for - there are many valuable resources exactly where you are.
What Washington is trying to say here is that he wants both black and white people to work together and rebuild the South in order to make it prosperous and forget about their differences.