If you have a mirror above your bed, the first thing you see each morning might be your own face. Once you've seen that your nose is still there, you're ready to face the day!
When you face someone you are across from them, maybe looking them in the eye. If you make a funny face, the person across from you might laugh. Or, if the person is your mother, she might tell you to stop it unless you want your face to stay that way. If you want to “save face,” then it's important to you not to look foolish in front of others.