time). The approach controller assigned me a transponder code and
told me "after departure remain clear of the class C airspace".

Lol - in my experience, either the airspace is too busy - or you sound like
you could be a nuisance.

. Well, the controller then called
me by my tail number and asked some questions (I don't remember
exactly what -- it might have been my expected cruising altitude and
aircraft type,

If you didn't automatically give him all that right after establishing
contact with him (assuming him), then maybe that's why he didn't want to be
bothered by you. I can just hear the controllers sigh when someone gets on
and says "podunk approach, november 12345, would like flight following" and
then the game of 20 questions starts.

to be sure that I understand this for the future. So ... if I'm told
to remain clear in the future, WHEN does that end?

When you explicitly get permission to enter.