An old flame is somebody you've been in love with for a long time. (You're actually the one whose heart is in flames) All that goes on in this dialogue is that Tilly now belongs to some other guy, the woman asks him how he feels about his big love being with somebody else and he replies that Tilly isn't his big love.

old flame refers to a person to whom you had a relationship with and the relationship is gone. This is an idiomatic expression. What is this in Chinese?

A "flame" is a slang word used as "boy friend" if you are a girl, or "girl friend" if you are a boy. An "old flame" means he used to be your boy friend, but not any more. If she was an "old flame" that could mean she was his girl friend for a long time, but not now. Any way, he lost her because she fell in love with another man. However, the man seems to deny that she was ever his girl friend. Maybe they were just platonic friends.... good friends but not romantic friends.