The poem seems to be spoken by a black man, an idea which is introduced by the direct declaration that "I am the darker brother." This, in conjunction with the opening line "I, too, sing America," suggests that there is another person he is comparing himself to, the person who already sings America and is perhaps a "lighter" brother. The persona narrating the poem is ambiguous because rather than pinning down a single individual, it could stand for any black man in America.
The following stanzas address the inequality. At the moment the speaker is narrating, he says that he is forced to eat in the kitchen when company comes. Tomorrow—perhaps not the literal tomorrow but rather a poetic indication of a day in the future quickly approaching—he will eat at the table. This reads as a metaphor for equality.
I have linked you to two websites. The first is the version of the poem I read. The second is an article published by the Smithsonian that talks about this poem. Take a look at the comparison between Hughes's poem and the poem by Walt Whitman. Whitman writes about all the different people who sing America, or in other words, those who celebrate the ideas represented by the concept of America as a democracy filled with opportunity. Hughes wants to add African Americans to that depiction by giving them the same freedom and rights as all other Americans.