My father always uses a small piece of bread to gather and scoop up the sauce remaining on his plate. As a child, I was taught to do the same, out of religious respect for the food, magnified in my parents by their experience of wartime food scarcity.
The expression usually refers to the tomato sauce left on a bowl after eating a portion of pasta, but as a child I had to fare la scarpetta no matter what dish I had eaten. The family tradition is now carried on by my husband, to my father's delight.
Scarpetta means little shoe, like the one lost by Cinderella, and the connection with the action described by the expression is unknown to me. What is known, however, is that fare la scarpetta is not acceptable during a formal dinner.