"Battle Royal" is part of Ralph Ellison's novel Invisible Man in which Ellison likens being black to being invisible to white race. There are several themes running through this short story, but perhaps the two most important are 1) a very tragic version of the "coming of age" experience and 2) the realization that personal accomplishments mean nothing for a black person in a society dominated by racism.
At the beginning of the story, the narrator is happily anticipating the speech he has been asked to give in front of the leading (white) citizens of the town. In this sense, he is a typical naif, that is, he is too inexperienced to understand his true position in this society. Because he is to give his speech in front of the most important people in the town--political leaders, religious leaders, school leaders--he assumes this affair is going to be a dignified and positive recognition of his achievements, but after he arrives at the meeting, he discovers that he is part of the Battle Royal: "I suspected that fighting a battle royal might detract from the dignity of my speech." As he will learn, unfortunately, dignity is the last thing the white "leading citizens" are interested in.
In an incredibly demeaning sequence, he and the other teenagers are forced to watch a naked white woman dance, threatened by the white men if they look away. This is, of course, an incredibly dangerous time for these black teenagers because they are set up to violate perhaps the most serious taboo in a racist world--being sexually attracted to a white woman.
After this horrific experience, they are forced to fight each other, and if they don't fight as the whites think they should, they begin hearing threats from the audience like "I want to get at that ginger colored nigger," an indication that these white men, despite the fact that they are the town's leading citizens, think nothing of killing a black. Even when the narrator attempts to bribe his black opponent to "throw" the fight between them, his opponent, behaving exactly the way the white audience expects him to, attacks the narrator with renewed vigor. He understands, of course, that the only safe way out of this situation is to behave as he is expected to behave.
The final insult in the fight sequence consists of the white men tossing fake coins onto an electrified mat so that can watch the teenagers react to the electricity as they attempt to pick up coins. The narrator, at this point, fully understands the he is a puppet in the hands of some perverted puppeteers.
By this point in the story, the narrator has come of age or, more to the point, come of race. He has, however, one more important fact to learn about his place in this society. When he finally gives his speech and mentions equality of the races, the reaction of the white men is, not surprisingly, negative. They make sure that the narrator understands that equality is not the appropriate goal--the appropriate goal is to know one's place in this society, and that place is not equal--"you've got to know your place at all times." The narrator finally understands his grandfather's dictum: "Keep this nigger-boy running."