The speaker of this poem is trying to answer the question: "What happens to a dream deferred?" (line 1). Deferred means delayed or withheld. Typically, if one defers something, one puts it off until later. Therefore, the question is really asking what happens to a dream if it is delayed or put off.
The speaker then offers a number of possibilities or outcomes of this dream; they are all similes until the very last one, which is a metaphor. First, he wonders if it dries up "like a raisin in the sun" (3). Next, he considers the possibility that it will fester and run (with pus), like an untreated sore. Next, he wonders if it will be comparable to food gone bad: will it "stink like rotten meat" or become inedible and crusted with sugar like a "syrupy sweet" (6, 8). In the final simile, the speaker considers that the dream might simply weigh a person down, sagging "like a heavy load" (10). Finally, in the only metaphor of the poem, the speaker asks, "Or does it explode?" Using a metaphor here, as well as italicizing the line, makes it seem as though this is the one the speaker considers to be the most truthful. It is certainly the most dangerous. While stinky meat or inedible raisins, or even a heavy load, might be inconvenient, they aren't terrible.
However, what does explode? A bomb: the speaker compares the "dream deferred" to an explosive. And bombs aren't simply inconvenient; they don't put out just one person, they can damage a great many people. Thus, the speaker seems to conclude that forcing someone to defer his or her dream will ultimately impact not just that person but many people, that doing so could actually be quite damaging for many.