I would say that the power of the speech lies in how King openly seeks to change the world. He does not shy away from saying this. He sincerely believes that his ideas of displaying moral fortitude in the times of crises can change both the individual and their societies. I think that the real power of the speech is how he aims the message to both the individual transformation as well as the social evolution that carries change as part of his message. The idea of loving "the enemy" comes first from the notion of loving thyself in King's mind:
Now first let us deal with this question, which is the practical question: How do you go about loving your enemies? I think the first thing is this: In order to love your enemies, you must begin by analyzing self. And I'm sure that seems strange to you, that I start out telling you this morning that you love your enemies by beginning with a look at self. It seems to me that that is the first and foremost way to come to an adequate discovery to the how of this situation.
For those who are enduring the worst in social abuse and mistreatment, I find it extremely powerful and compelling for the argument to be flipped on the individual. The notion of self- reflection is effective because it seeks to make the individual a priority. In a setting where the person of color is continually denied dignity and self worth, Dr. King sought to bring these elements back into the equation. In doing so, Dr. King was able to bring power back to the person of color in the South. Simply put, how awesome is that? I think that this is where his speech is so effective. Unlike other Civil Rights thinkers who continually argued that the problem was in the hands of others- White government, White society, White racists- King flips the script and says that if individuals can demonstrate the capacity for loving themselves in a pure and authentic manner that transcends contingency, then they are able to reassert power which disarms their enemy. The notion of "loving thy enemy" becomes a profound means of social and personal change, enabling a radical notion of changing the world to become evident.
I think that this is where the speech gains its power. In echoing the words and thoughts of Gandhi, Dr. King is not merely talking about the Civil Rights Struggle. He is speaking about the "civil war" that exists in "all of us." This is effective because he is arguing that people of color must liberate themselves first before any political liberation can happen. In this, Dr. King's message is similar to that of Christ, appealing to a true notion of Christianity in its application for individual, social, and political transformation.