Apologies in advance to the chemists in the room, because I'm going to butcher the science on this. But the lay explanation is fascinating.
Weight loss discussions typically focus on two pathways, or both in combination: caloric restriction (i.e. eating less) and exercise. In both cases, the goal is to "burn" more calories than we take in and, thus, remove excess fat. But what does this mean in practice? Calories are a measure of heat energy, so the term "burn" seems to make intuitive sense. But the theory of conservation of mass tells us that mass cannot be created or destroyed. We are not losing weight through heat.
If the common wisdom is a lie, the next idea is that we lose weight through digestive excretions, i.e. feces. But this, also, is incorrect, for somewhat obvious reasons. The digestive system is concerned with taking in fuel, breaking it down, using it, and getting rid of all the useless bits out the other side. Nowhere in that system is there any "burning" or converting of stored energy. In short, we don't lose weight through our poop.
Losing weight actually comes down to metabolizing triglycerides, the primary component of fat. Triglycerides are essentially a bunch of carbon and hydrogen with a bit of oxygen thrown in. This is basic chemistry, and I have forgotten most of my chemistry. But wait, carbon? Hydrogen?
So, it turns out that the vast majority of "burned" calories are expelled through breathing. Eighty-six percent, to be precise. How? Well, just how we were taught in elementary school — O2 in, CO2 out! Most of the remainder, i.e. those hydrogen atoms, leaves as water, H2O coming out of all the various places that we get rid of water, such as sweat, spit, tears, and urine.
Hearing this for the first time, it seems utterly crazy. But actually it makes a lot more sense than the idea that all that fat is being magically "burned" away.