As in using “take care” as a sign off or “goodbye”?

It’s a shortened version of “take care of yourself”.

Take care not to suck any dick on your way through the parking lot.

Hey, get back here!

It’s a phatic expression so it doesn’t really mean much. It’s only there to signal the speaker acknowledges the end of the interaction between them and the hearer, in a non-hostile way.

I’m not sure if the impersonal tone is there to convey social distance between the speaker and the hearer, or if it became like this due to overuse.

I think its kind of a nice sentiment to say “take care [of yourself]”, but recently ive been saying “stay safe” when I want to slightly stress people out.

Use caution. Spend some time thinking about what you are doing before, while you are, and after doing something.

It’s an expression from a different age that has mostly retained its meaning while its components “take” and “care” have undergone semantic shift.

The word “take” originates from Middle English “taken,” which comes from late Old English “tacan,” meaning “to grip or seize.”

The word “care” originates from the Old English “caru” or “cearu,” meaning sorrow, anxiety, or grief, specifically the type by which one person has concern and takes responsibility for the situation.

So “take care” could be translated as “seize empathetic responsibility” and is a way of showing the speaker’s wish that the other person behaves in a socially responsible manner.