The table in the dining room is where important people meet, talk and dine as equals. Right now, the speaker, being black, is forced to eat in the kitchen. As a descendant of slaves, he is not deemed a human being who has his own will and the right to say something. He is not even deemed a person who has every right to love America and try to change it for the better.
In this poem, he reclaims his right to metaphorically sit at the table, which means to be treated as equal to the white people.
