What does King mean when he says that nonviolence “cuts without wounding?”

What King is saying here is that nonviolent resistance is
effective in making change possible, but that it does not create anger or
hurt.

Martin Luther King, Jr. was a disciple of Mahatma Gandhi and
Henry David Thoreau, two of the main proponents of the idea that people should resist injustice
by nonviolent means. King believed that nonviolent resistance was both more effective and more
ethical than violent resistance. This view is shown in the statement that you cite. King believes
that nonviolence "cuts" (it works, it is effective) but yet is ethical because it does not bring
harm to those it is directed at (it does not "wound"). This is because it treats them with
respect and does not give them any reason to feel angry at those who are
resisting.