The Question and Answer section for The Glass Menagerie is a great resource to ask questions, find answers, and discuss the novel.
Tom is the breadwinner and caretaker of his family. Amanda's husband abandoned the family long before, and Tom has taken on the role of man of the house.
Amanda tells Tom that he needs to find someone to take his place in the family.... a husband for his sister.
Tom stays with his mother and sister because he feels obligated. He's the man of the house... he feels as if it is his job to stay and care for his mother and sister.