What lessons does "The Possibility of Evil" teach about human nature?
One lesson the story teaches is that (some) people are duplicitous. In other words, they are "two-faced." Miss Strangeworth is kind to everyone's face but as soon as they are out of earshot, she is critical and judgmental. She plays the role of the amiable neighbor in public. But in private, she is condescending and, at times, even spiteful and hateful towards others.
This duplicity shows Miss Strangeworth's feelings of superiority. She feels that it is her duty to send these anonymous letters. She thinks that she is somehow more righteous, ethical, and logical and therefore in a position to judge other people. She has a warped Messianic complex. The fact that she does this anonymously shows cowardice, but Miss Strangeworth is so conceited that she probably doesn't even consider this to be a cowardly practice.
So, even someone like Miss Strangeworth, who seems like a good person on the surface, is capable of evil. The title suggests the possibility of evil in anyone and particularly, in people like Miss Strangeworth. She is not really helping these people because she is not giving constructive criticism. She is simply being insulting and condescending.
Her use of anonymous criticism is somewhat similar to the hate speech and ugly comments we see on the internet today. Going under an anonymous screen name gives some people the notion that they can say anything, with no consequences. It is easier to be self-righteous and judgmental when one is removed from those whom she/he is criticizing.
This story shows negative aspects of human nature: duplicity, self-righteousness, and cowardice.