We're guessing that the "darker" here refers to skin color. Hughes wrote a lot about the African-American experience, so he's probably talking about black Americans. (Check out, for example, "The N**** Speaks of Rivers" and "Harlem.") Not about "darker" things like, say, Voldemort.
But back to brotherhood. From the previous line referencing America, and knowing what we (now) do about the reference to Walt Whitman's poem ("I Hear America Singing"), we can probably go ahead and say that Hughes is referencing the "brotherhood" of all men in America.
So he's asserting his identity not only as a black man, but also as a vital and familial part of American society as a whole.
Notice also that it says "the darker brother" and not something like "one of the darker brothers." Is he speaking for all of the black community in America here? Kind of sounds like it to us.
They send me to eat in the kitchen When company comes,
Now, we're pretty sure that the speaker of the poem isn't literally being shunted into an actual kitchen the moment someone arrives. So what's Hughes talking about here? And who's "they," anyway?
Well, what this is, we think, is an allusion to the days of slavery – when the house servants were confined to their quarters when guests came to the house, to keep the slaves (and, by association, their race) out of sight.
Of course, though slavery had ended by Hughes's own time, racial segregation was still very much alive and well, so these lines also remain completely relevant to when Hughes himself was writing. These lines might also allude to a 20th-century house with black servants. There are all kinds of layers of time and meaning here.
What's wrong with eating in the kitchen, you ask? Well, it isn't like eating in the dining room as a member of the family, now is it?
The "they," then, becomes pretty clear – it's white people.
This bit, however short, paints a pretty dismal picture of a (not all that distant, mind you) past in which blacks and whites were segregated, with blacks being thought of as inferior.
But I laugh, And eat well, And grow strong.
Just after we get two lines that bring the entirety of slavery to bear on the poem (making it a weighty affair indeed), we have the speaker just laughing, eating, and growing strong.
We also want to point out once again that the "I" in this poem is a kind of plural "I." What we mean by this is that the speaker here isn't actually speaking just for himself – he's speaking on behalf of his entire race and the history of that race.
So even though our speaker's been sent away while company's around because of his race, his appetite isn't any worse off for it. Nor is his sense of humor. We can almost get a picture here of the speaker and the other house servants having their own little dinner party in the kitchen, growing strong with each other's support. And having a heck of a time.
These little lines set up the beginning of this poem's "turn," which is a kind of poetic change of mood.