The phrase doesn't mean actual writing, but that the feeling is so clear it might as well be. When she won the award, joy was written all over her face.
Pro Tip 2/3
Negative Emotions
It's often used to describe negative emotions that someone tries to hide, but fails. Even though he pretended to be fine, his sadness was written all over his face.
Pro Tip 3/3
Positive Emotions
It can also highlight positive feelings that are unmistakably displayed. Her pride in her children's achievements was written all over her face.