In the paragraph preceding the one in which this quote appears, King mentions that, like the Biblical apostle Paul, he feels compelled to go wherever there is need of his help. He then states that all people in the United States are tied together in a web of mutuality in which an injustice in one place threatens justice anywhere.
This means that once people of ill intent see a group get away with an injustice in one part of the country, they are emboldened to try it somewhere else. The more locations in which people, for whatever reason, get away with further injustice, the more easily it can spread. It can even start to seem normative.
Although he keeps his focus on the United States and doesn't specifically quote John Donne, Donne's words echo behind what King says. Donne too was a clergyman, and in a famous meditation he noted that we, as human beings, are all interconnected. When a bell tolls to communicate that someone has died, it effects all of us, even if we didn't know the person, because all people contribute their gifts to the world.
King wants his audience of mostly white pastors sitting on the sidelines of the Civil Rights struggle to understand that blacks, even in a far away place, are an important part of the fabric of American life. He is urging them to join in the struggle, saying it is about them too.