This idiom or expression has to do with:

Helping, as in:

I’m tired of bailing you out of your problems. Next time call someone else!

Also, related:
Helping someone out of jail by paying the bail money.

This idiom or expression has to do with:

Saving their job, life, reputation, etc.
Saving them from difficulty, exposure, embarrassment, etc.

A. You know, when you took the responsibility for wrecking the car, you really saved my skin!
B. It’s okay. That’s what friends are for!

Also:
Saving someone’s hide, or neck, or bacon.

This idiom or expression has to do with:

Getting or giving help.
Getting or giving a head start.

A. You think he can do the report on his own?
B. Yes, but if you give him a leg up I would really appreciate it.