The states Dr. King mentions in his speech are being cited as representatives of the many places in which racial discrimination of any sort exists. He is recognizing the ways in which equal rights still don't include African-Americans, and he is encouraging those listening to his speech to return to those places and continue the fight to bring about true equality for all people.
He highlights voting rights, contrasting situations in the south where blacks were not allowed to vote with situations in the north where blacks could vote but there were no candidates for office who would have stood up for their viewpoint if elected to office.
We can never be satisfied as long as a Negro in Mississippi cannot vote and a Negro in New York believes he has nothing for which to vote.
He recognizes those who had been involved in demonstrations and marches and other events in the southern states. In spite of having been beaten by the police and drenched with fire hoses and choked with tear gas and taken to jail after being arrested, Dr. King urges his listeners to
Go back to Mississippi, go back to Alabama, go back to Georgia, go back to Louisiana, go back to the slums and ghettos of our northern cities, knowing that somehow this situation can and will be changed.
The states named were not the only places in which such events occurred, but they were possibly the most publicized locations. As such, they earned particular and specific mention in his speech, on behalf of all the places in which the work of realizing Dr. King's dream was not yet finished.