This speech is often called the "Atlanta Compromise" because Washington encouraged African-Americans to worry less about social segregation and more about achieving economic security. But it was mainly aimed at whites, and he used lots of imagery, most famously separate fingers on the same hand, to suggest to his white audience that blacks did not seek to upset their social superiority, and that the races could co-exist. The speech was essentially an acceptance, if not an endorsement, of Jim Crow, which received the sanction of the courts one year later in Plessy v. Ferguson. The speech represented a gradualist, some might say accomodationist, approach to the race problem in the South. The most outspoken critic of Washington was W.E.B. DuBois, who accused him of aiding and abetting a "national crime."
Basically, what Booker T. Washington wanted was for his fellow African Americans to be patient and not to be pushy. He wanted them to work hard at their jobs so that white people would respect them. He believed that, if they did that, the white people would give them rights.
It was in this speech that Washington advised African Americans to "cast down your buckets where you are." In other words, he wanted them to accept their current place in society and work hard in that place. He believed that the white people would respond well to this sort of deference on the part of the black community and reward it with good treatment and, eventually, equality.