=to leave a place/event without saying “good-bye” or without permission. Why? The reasons can be different, for instance, not to upset the host.
Curiously, in many other languages, e.g. Russian, Italian, Hungarian, Polish, etc, this phrase sounds like “leave English style”.
TAKE FRENCH LEAVE is also used when sb leaves work without asking for permission:
I’m exhausted. This week I’ve been working much more hours. I think I’ll take French leave and go home.
It can be used in a military context (to be absent from a military unit):
Some soldiers took French leave and were punished for this.