The narrator dreams at the story's end about attending the circus with his grandfather who "refused to laugh at the clowns." Later the old man tells him to open his briefcase and read what's inside; the narrator find an envelope with an endless number of other envelopes inside to open. Finally he opens a different envelope with "a short message in letters of gold": "To Whom It May Concern, Keep This N--- Boy Running."
After this dream narrator remembered for years, he "had no insight into its meaning. First I had to attend college." So he has no immediate epiphany. However, he discovers that the "endless" envelopes refer to the many times he will be deceived by others; for instance, we have already seen that he was unaware he would have to fight others before he gave his speech. Like the other boys, he was tricked by the fake money on the electrified rug.
This story is Chapter One in Ellison's novel Invisible Man. The narrator will indeed be kept "running" by other people, mostly white, who trick and mislead him. Even in a dream, the grandfather predicts this future for him, reinforcing his death bed remarks. The narrator will, as amy-lepore's answer indicates, be reminded of his grandfather's words and this dream at key points throughout the novel before he finally grasps modern society's view of him as an insignificant "invisible" man.
The narrator has the dream with his grandfather on his deathbed saying, "Kill'em with smiles. Yes'em to death." The epiphany here is that the grandfather realizes and the narrator learns that nothing is given to the black man for his talents. The white man doesn't care about what you can do, what he focuses on is how well you behave and how well you say, "Yes Sir." So, in essence, the grandfather is telling him to say one thing while doing your own thing in order to keep your sanity and your dignity. This is a turning point in the novel, and he remembers this deathbed declaration every time he is challenged in the novel--with the blonde woman and the battle royal, with the driving of the white benefactor at school, with the experiences in New York, until finally we find him living alone with hundreds of lightbulbs burning and he babbles about "sticking it to the man".