Why does Miss Strangeworth write the letters?

Miss Strangeworth has developed some very distorted thinking and an unhealthy proprietary interest in a place "she had never spent more than a day outside." She believes that "there wouldn't have been a town here at all if it hadn't been for my grandfather and the lumber mill."
Because her entire life of seventy-one years has been spent in the same place, her range of experience is quite narrow. She is a reactionary who believes that the values she has embraced all her life must continue, and her letter-writing campaign is a pathological exercise in attempting to force her beliefs on others. The rigidity of her habits is demonstrated when she goes to Mr. Lewis's grocery and chides him for "forgetting that I always buy my tea on Tuesday." She is so self-absorbed that she doesn't stop to consider that Mr. Lewis has responsibilities beyond catering to her rituals—or that others have the right to live their lives as they see fit.
The hateful letters that Miss Strangeworth writes are an outward manifestation of her self-righteousness. It cannot be reasonably said that her letters are meant as constructive criticism. Instead, they are deliberately hurtful and do not recommend any corrective action. Because Miss Strangeworth's life is empty, it allows her plenty of time to observe and sit in judgment of others.

Miss Strangeworth in "The Possibility Of Evil" writes poison-pen letters to various people in the town. She does this because she believes herself to be the town's self-appointed guardian of public morals and standards. She also acts like she owns the town; her grandfather built one of the first houses on Pleasant St, so she has deep family connections to the place. She believes they should build a statue to him. Miss Strangeworth has a very clear notion of how things should be run and how people in the town should behave.
On her daily rounds, she regularly dispenses advice (whether asked for it or not) to anyone who will listen. She has a particular view about how parents should raise their children. This is reflected in one of her nasty letters:

Didn't you ever see an idiot child before? Some people just shouldn't have children, should they?

Miss Strangeworth doesn't see her behavior as in any way unacceptable. As far as she's concerned, she is simply alerting people to the dangers of unchecked evil. Even when someone returns the favor and sends her a poison-pen letter, she doesn't feel in any way chastened; she simply feels, more than ever, the never-ending wickedness of the world.