How do calories leave your body when you exercise?

Your cells use fuel from food.  Calories (or really KCals) are simply a way we measure the potential energy/fuel available in any type of food.  The more energy your cells need to do the work they are doing (sleeping vs running up a flight of stairs), the more fuel they will use.  This translates into calories burned.  The leftovers from the used fuel are excreted mostly via your kidneys as urine but can be sweat.  However, that isn't the same thing as calories bursting out of your skin.  

A calorie or kilo-calorie is a measure of energy, which is the capability to do work or cause a change. Calories are nothing that leaves you body as you exercise. They are something you use, the energy, to do those exercises.

But when you do exercise and use up calories doing it, not all that energy is used in exercising. Some of it is in fact lost and not used at all. And the lost calories are typically in the form of heat, which as your "someone" suggested is lost to the air through the sweat.

It is used as ATP because it is not calories, it is broken down into different nutrients, etc.

burned-up in the muscles making energy ...............................................