This is compiled from my own conclusions and some read on the aforementioned discussion about the Chicago show (google The Minutes Letts spoilers)
The Mayor's words got to Peel. He initially fought back against the accusation that he would forego his ideals in favor of comfort - this is his immediate reaction and thus exit into the rain.
Heavy rain serves to wash things away, and I think this is what prompted Peel's 'wake up call'. He had no recourse that would lead to a better world for his daughter. The changemakers and ones with power were in the room he had just left. And the rain, literally and metaphorically, would wash away any blood on their hands for actions they took to preserve the future they wanted to see.
So he goes back in. He acknowledges that he would bathe in the blood of past atrocities to affect his family's future. He's surprised that their routine is, in effect, an appropriation of a Native American dance, showing once more that the council will continue to take everything for their own in the name of Big Cherry, but he eventually begins to pick it up.
My thoughts. Would love to hear other people's interpretations.