Ash in a food is anything that is not a nutrient we directly use for energy. It's basically all the minerals such as calcium, potassium, etc. Not like campfire ash, don't worry.

It makes sense when you think about it this way: if you incinerate a sample of food, all the energy nutrients (proteins, fats, carbs) and water content burn away, leaving behind the noncombustible minerals. Since it is left behind by the fire, we call it ash.

Calories are counted by literally burning food and counting how much energy is created from the fire. "Ash" is probably literal ashes.