During these chapters, it seems to me there is really only one way Jem and Scout make Cunningham stand in Atticus' shoes. Scout and Jem interrupt a late night event between Mr. Cunningham and Atticus. The event is a sort of altercation that Mr. Cunningham wants to unleash on Atticus. But, because Scout demonstrates her natural and sincere childhood to Mr. Cunningham by talking about Mr. Cunningham's past problems (which remind Cunningham of Atticus' prior help), and talking about his own child, she helps Cunningham stand in Atticus' shoes. She makes Cunningham realize Atticus only does what he does to provide for his children. This is the dream of every father.
Cunningham may very well have shared this experience with one of his relatives because by chapter 22 and 23 we learn that someone with Cunningham background was on the jury. Atticus puts together the fact that someone also took considerable wearing down on the jury. Thus, Atticus relates their experience at the jail that night as noteworthy.