How old is Miss Brill? What are her circumstances? Why does she listen in on conversations?

Miss Brill is likely fairly elderly.  She seems not to be in the best of health, as after walking to the park, "She felt a tingling in her hands and arms [...]."  This is not a typical sensation one has after a leisurely walk; it sounds a lot like neuropathy, a condition associated with the elderly.  Moreover, when the young couple sit down next to her, the boy calls her a "stupid old thing" and expresses his wish that she would "keep her silly old mug at home."  The commonality between these two comments is the word "old."  Earlier, Miss Brill had come to a realization as she looked around at people in the park: "They were odd, silent, nearly all old, and from the way they stared they looked as though they'd just come from dark little rooms or even -- even cupboards!"  Then, at the end of the story, she returns to "the little dark room -- her room like a cupboard."  This similarity in the descriptions of their rooms and her own lets us know that Miss Brill really is old and that she just never realized it until now.

Miss Brill does not have much going on in her life.  She lives alone and only gets out four times a week to read the newspaper to an "old invalid gentleman [...] while he slept in the garden" and once on Sundays to come to the park when the band plays.  The majority of her interior life seems to be imaginary.  She concocts an elaborate fantasy about how everyone at the park is in a play -- herself included -- and that she is actually an important actress.  "No doubt somebody would have noticed if she hadn't been there; she was part of the performance after all."  Perhaps she secretly fears that she wouldn't be missed if she stopped coming, and so she invents a fantasy wherein she plays a vital role.  Further, she looks forward to telling her old invalid gentleman that he is having the newspaper read to him by an actress!

This is, also, the reason she eavesdrops on others' conversations.  There is so little of interest in her own life that "She had become really quite expert, she thought, at listening as though she didn't listen, at sitting in other people's lives just for a minute while they talked round her."  Again, however, she doesn't realize that this is why she eavesdrops, at least not until the very end of the story.  At that point, she hears something crying, and readers realize that it must be her.