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.