In the phrase you cite, Dr. King is introducing the first in a series of metaphoric descriptions of the circumstances in which Negroes in the United States found themselves living in 1963. His purpose in all the descriptive phrases is to highlight the isolation and deprivation faced by Negroes.
Dr. King compares Negroes in 1963 to slaves of one hundred years earlier. He claims that they are still shackled, not by manacles and chains as were the slaves of the 1860s but by segregation and discrimination. He does this to create the awareness that slavery is not a condition from the pages of history but is still present at the time of his speech.