First of all, it’s a poem. With a lot of questions. Its central question, however, is "what happens to a dream deferred?" meaning basically "what happens when your dreams don’t come true?" only Hughes’s version sounds better. He poses a couple of graphic suggestions: do the dreams dry, fester, run, stink, crust, sugar, sag, or explode? If we make like kindergarteners and play a little game of, "what doesn’t belong," the word explode fits the bill. While the other verbs deal with decay, waste, or extinction, explode is considerably more active, more powerful. No wonder Hughes chose to use it last.
All right, time to explore some context. Hughes was a major leader and poet in the Harlem Renaissance, which means that although anyone can have dreams, Hughes was specifically addressing the situation of blacks in America, who had been systematically denied access to the various American dreams of education, career, purchasing power, etc. Asking if deferred dreams explode is a subtle (or not so subtle) way of reminding readers that deferred dreams don’t always decay and disappear; they can very well trigger explosions.
So why did Hansberry use this as the epigraph for her play? Although the play ends with the Younger family achieving one of their dreams – of moving out of the slums and into an actual house – the play leaves us hanging in regards to the other dreams. Beneatha still needs tuition money for medical school and Walter still doesn’t have a rewarding job. This reflects the similarly in-between situation of blacks in America at the time. While they had fulfilled some dreams (i.e., freedom, integrated education) there were still plenty of ones that had been continually deferred (i.e., equal opportunity employment). The epigraph is a way for Hansberry to point to both the universal nature of her play – everyone has dreams – and its particular nature – black Americans have been forced to defer their dreams more than others.