Ritual/symbolism of putting teapot in cup

In Eat, drink, man, woman (Yin shi nan nu, 1994), at 1:44, one of the daughters pours hot water out of a large pot into a tiny one, then pours the tea out of the tiny pot, then puts that one in the cup:

She is in a teahouse, having a conversation with the man she thought a love affair was developing with, but who (the day before) turned out to be married already:

Thanks for coming. I thought you might not show up.
Why not?
The other night...
... (interrupted by waitress)
(First pouring)
I understand what you've been going through... although I can't explain it.
Neither can I...
It's complicated, but I can understand that.
Thanks.
(Second pouring and putting the small pot in)

I can't help thinking that this is a ritual and/or symbolic for what's going on.
Can anyone explain the symbolim of this last gesture?