When Martin Luther King, Jr., named so many separate states in his speech, he not only connected with his audience, but he also subtly advocated for a federal solution to the problem of racial discrimination. The setting of the speech was at a March on Washington that drew more than 200,000 people, the largest gathering to ever convene in Washington, D.C., at that time (1963). People from all around the country had made an effort to attend the March, particularly civil rights activists from Southern states. He specifically addresses those attendees when he says, "Go back to Alabama, go back to South Carolina, go back to Georgia, go back to Louisiana" with hope in their hearts that change is coming. Not only that, but legislators and sympathizers, both black and white, also attended, and by naming some northern states later, such as New Hampshire and New York, and the western state of California, he acknowledged that people from all around the country were listening to him and supporting the cause of racial equality.
By naming such a wide variety of states in his speech, one could say that Martin Luther King, Jr., was trying to "make a federal case" out of racial discrimination. That saying means exaggerating something trivial, which his opponents in the South no doubt believed he was doing. But to take the phrase literally, because some Southern states had implemented Jim Crow laws with impunity, the solution seemed to be to get the federal government involved to outlaw such discriminatory practices and improve the possibility of black people rising to political positions in those states. Ultimately the Voting Rights Act of 1965 was a national solution to the problems posed by certain Southern states that had the most egregious anti-black policies. Naming not only the offending states but also the friendlier states was a way to make the problems black people faced a national issue, not just a state issue.