I think the author sees the aunt as rather ineffective at entertaining the children. All the kids are fairly small and their aunt is unable to capture their interest. She has trouble answering their questions regarding their surroundings, so she decides to tell them a conventional moral story that fails to entertain them. When a unknown man points out the flaws of her story, she challenges him to tell a better one. The children do indeed enjoy the man's story more, however the aunt finds it improper. The aunt is a rather flat character, posing as a conventional authority figure.
Most of the aunt's remarks seemed to begin with "Don't," and nearly all of the children's remarks began with "Why?" The bachelor said nothing out loud. "Don't, Cyril, don't," exclaimed the aunt, as the small boy began smacking the cushions of the seat, producing a cloud of dust at each blow.