How does Miss Strangeworth's letter-writing help the reader understand about what kind of person she is?

Miss Strangeworth seems like a nice, simple little old lady in the opening part of the story. But she turns out to be a complicated character. The three letters she writes when she gets home have one thing in common. They betray the fact that she is unconsciously filled with envy at people who have something in their lives that has always been missing in hers. She tells herself that writing her letters is fulfilling a civic duty, warning people against the possibility of something bad happening to them if they are not watchful. But in each of the three letters we actually read, we can see that she has a motive she is not even aware of herself.

The letter she writes to Don Crane strongly suggests that she envies him and his wife for having a baby they both adore. She has never been married and may have always dreamt of having a baby of her own. Of course, it is too late now. So she warns Don against having any more babies. She writes: 

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

Her letter to Mrs. Harper, which follows up on others she has previously mailed to her, suggests that Mr. Harper may be having an illicit affair with another woman who lives right in their town. Miss Strangeworth has never had a husband. She can at least try to poison the marriage of a woman who does. She writes:

Have you found out yet what they were all laughing about after you left the bridge club on Thursday? Or is the wife really the last one to know?

We do not see the letter Miss Strangeworth has written to Linda Stewart's parents, but we can see that Linda and Dave Harris might have been the target of her secret envy because they are so young and so in love.

The letter that goes to old Mrs. Foster seems intended to arouse fears about of her pending operation as well as to poison the relationship between Mrs. Foster and her nephew. Miss Strangeworth writes:

You never know about doctors. Remember they’re only human and need money like the rest of us. Suppose the knife slipped accidentally. Would Doctor Burns get his fee and a little extra from that nephew of yours?

This letter is subtle and complex. It suggests that Mrs. Foster must be financially well off, while at the same time it suggests that Miss Strangeworth is not. Miss Strangeworth has two reasons for envying this woman. Mrs. Foster is wealthy and has a nephew who cares about her and looks after her. Mrs. Strangeworth has nobody, and never had anybody.

Miss Strangeworth's anonymous letters are commonly called poison-pen letters. In all the cases mentioned in the story, her letters have the effect of poisoning someone's relationship with another person. In the case of Don and Helen Crane, the letter about their baby could easily hurt their relationship by making them afraid to have any more children. 

Miss Strangeworth is perhaps more to be pitied than censured. She is a lonely and unhappy old woman leading an empty life. She resembles Emily Grierson in William Faulkner's story "A Rose for Emily" and Miss Havisham in Charles Dickens' novel Great Expectations.