There are many metaphors in this speech: King's way of speaking was beautifully picturesque, drawing comparisons and images wherever he could. Many of the sentences in this speech contain the same simple metaphoric structure, where one noun is compared directly to another noun, using the word "of." Take an early paragraph in the speech as an example:
There will be neither rest nor tranquility in America until the Negro is granted his citizenship rights. The whirlwinds of revolt will continue to shake the foundations of our nation until the bright day of justice emerges.
This paragraph uses this characteristic, simple structure twice: "the whirlwinds of revolt" and "the bright day of justice." These are metaphors in their most rudimentary form: they serve to have one noun directly take the place of another in the sentence. This allows King to create a more descriptive and moving picture of racial injustice and the fight against it. He does not just describe a "revolt," but metaphorically calls revolts "whirlwinds," evoking the constant effort and powerful impact of those participating in revolt; they are always spinning, moving, working, effecting change. The metaphor of "whirlwinds" also describes well how the revolt against persecution and injustice is meant to disrupt "rest and tranquility" in America, just as winds disrupt rest and tranquility on a still day.
Similarly, the metaphor in "the bright day of justice" allows King to better describe what justice will be like. Justice is a "bright day," full of possibility and happiness. Like a bright day, justice doesn't come only to a few people, but shines equally over the whole world. The two metaphors interact with each other: whirlwinds blow through to result in clear, bright days, just as the effort of revolt results in justice. King uses metaphor in a simple but effective way: he replaces a literal word with a metaphorical one, and in doing so better describes his ideas, making the interaction between those ideas more complex, too.