The law of mass conservation says that during a chemical reaction, all the atoms you had at the beginning will still be there at the end; the only thing changed in the reaction is the way they're bonded together. That means that, even when you manage to trick your body into breaking down stored fat or whatever, those atoms from the fat molecules are still inside you even after the energy from their chemical bonds has been harvested. For you to actually lose weight, the broken-down stuff has to leave your body somehow. There are a only a couple of ways mass exits our bodies so it wasn't too hard to narrow this down, but the answer probably isn't the one you're thinking of.
You were thinking "you poop it out duh," weren't you? I was too, because poop is funny but also because it's the most obvious way that mass exits your body. If you think about it though, that doesn't work-- poop is what's left over after your digestive system has gotten everything useful that it can out of whatever it is you ate-- nothing from the body is in there, in other words, just undigested food and maybe some gut bacteria picked up along the way.
So how about pee? That works a little better; the stuff that eventually becomes urine is filtered from your bloodstream by your kidneys, so it was at least part of you at some point. The main function of the kidneys is filtering waste from your blood though; your urine is <95% water with very trace amounts of things you don't want in your bloodstream, none of which are the results of fat breakdown.
So what's left? Respiration, which if I'd been thinking about this like a chemist (not being a chemist, that was never too likely) would have been fairly obvious. When your body metabolizes fat, it's basically taking a fat molecule (which is a large carbohydrate) and oxidizing it. Any undergrad chem student can tell you that the results of oxidizing a carbohydrate are energy, water, and carbon dioxide. The chemical reaction looks like this:
So post-breakdown, the mass that was formerly stored fat is now excess CO2 and water in the bloodstream. Luckily, our body has very efficient ways of getting rid of both of those things. The water, which only comprises about a quarter of the final mass here, will in fact end up getting peed out, but the majority of the mass is removed when your lungs exhale carbon dioxide during a process known as "breathing." So basically once the fat's broken down, you just breathe it away. Learning this led me to the obvious conclusion-- I'll just hyperventilate to lose weight! Screw dieting!-- but then I remembered that your body has to break down the fat first for this to work.
It's not completely useless information though! One consequence of this is that, when we lose weight, we're adding CO2 to the atmosphere and directly contributing to climate change. In other words, you are destroying the world if you go on a diet. Why do you hate the environment, FitBit?