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Martin Luther King Jr. is making the point that, while the United States had prospered and become the wealthiest nation on Earth, the overwhelming majority of black people in America were poor.
The racial divide in 1963 America was very much also a wealth divide:
(Of course, one might argue that little has changed in that regard!) Segregation was, as I'm sure you are already aware, still very much the reality of life for millions of African Americans.
By using the analogy of lonely islands of poverty in a vast ocean of material prosperity, MLK was encouraging people to visualise that racial divide, and their segregated society, in those terms and to demand racial justice for mistreated African Americans.