The metaphor that King gives in this example is that of "horse and buggy pace," which refers to transportation before automobiles. The metaphor serves two purposes. First, it emphasizes how slow the advancement of civil rights has been for African Americans by presenting an image that any modern reader would associate with a slow pace.
Secondly, the metaphor also provides an image of the past. This is to highlight how civil rights advancement is being conducted in a way that does not resemble the modern era, but that is instead reminiscent of a time with less social progress.