In Chapter 15, why is Scout's statement, "...but you said...that you all'd ride it out together..." so powerful?

Scout is able to make Mr. Cunningham stand in Atticus's shoes for one moment.  By stating the line you listed in your question, Scout forces Mr. Cunningham to remember what Atticus did for him.  He is also forced to see Atticus as a man, and one who helped him out of a difficult situation, as well as a father, like Cunningham himself.  Atticus has become a person rather than just someone standing in the mob's way of a lynching.

This causes Cunningham to feel empathy and compassion for Atticus.  After all, Atticus is simply doing for Tom Robinson the same thing he did for Mr. Cunningham.  It is this realization that causes the tension to diffuse and Mr. Cunningham dismisses the mob.

Finally, this is an important point in the novel because it helps reinforce what Atticus has always been telling his kids - to crawl inside someone else's skin before judging them.  Scout offers Mr. Cunningham a glimpse through Atticus's perspective and that makes all the difference.