I've heard it's a reference to an obscure and unsuccessful Tennessee Williams play that premiered on Broadway in 1962. A wealthy woman finds a young man trespassing on her land, they engage in a lengthy series of dialogs, and at the end she dies of a terminal illness. The original Broadway production starred Tallulah Bankhead, who some said was far past her prime.
I keep looking for someone else to find something sexual in the lyrics -- Grace liked to slip clever things into her songs -- but no one has explained it to me to my satisfaction.