Lamentably, it is an historical fact that privileged groups seldom give up their privileges voluntarily.
Another good reason to read the letter in its entirety. The whole of the letter, including the paragraphs that @mint posted above, is about the necessity of nonviolent direct action. There isn't much reason to think that the historical fact described by King in this quote is any less true, today.
Speaking personally, it's also a provocative line for white allies and accomplices. It's worth asking myself, again: what privileges I am keeping close? If I'm not confronting and wielding that privilege to bring about a more just world, then what am I doing with it?