The location and timing of his speech was another thing that made it so effective. He gave his speech on the steps of the Lincoln Memorial, in front of the statue of the man who freed the slaves nearly one hundred years earlier, in part because of the majesty of the setting in the nation's capital, and near the White House, and in part because it had been almost a century since slavery ended, and he wanted to drive home the point that African-Americans had been waiting far too long for some semblance of equality in the United States.
His speech took place during the March on Washington in 1963, as hundreds of thousands of blacks and whites came to the capital in a show of support for a new Civil and Voting Rights Act.
King was also a master at speech delivery, and knew the "I have a Dream" line was the strongest of his speech, so he timed its delivery right at the point where it would get the largest response, and accentuate his main points perfectly. The fact that he was a southern Baptist preacher didn't hurt either and heavily influenced how he delivered his speeches. He also liked to introduce scripture into his speeches to add the weight of religious authority to his cause and motivate his supporters.
What Martin Luther King, Jr. was trying to do in the speech you mention was to set forth his vision of what the United States should be like. He was trying to persuade people that the US should become a society where race does not matter and where everyone can have an equal opportunity to succeed.
He did a number of things that were effective. First, he used a variety of rhetorical devices to make his speech more compelling and persuasive. An example of this is how he kept repeating "I have a dream..." Second, he used a great deal of religious imagery, which is popular in the US. Finally, he appealed to a basic idea in American thought -- the idea that all people are equal.