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 means:

You have to face life.
You can’t hide the truth.
You can’t run away from life.
You brought this upon yourself, now you have to face the consequence.