They can start with the moment of birth. For instance, right after a baby is born, the first diaper is put on. So the story could start with the hospital setting, the nurses or doctors handling the baby gently and wrapping the diaper around the tiny body.
Another way these stories start is when the baby is a bit older and has an unexpected accident while not wearing a diaper. Then the parents decide it's time to introduce the diaper, and the story kicks off with that decision - perhaps a trip to the store to pick out the perfect first diaper, with all the different options and choices they had to make.