There are numerous theories, myths and ideas surrounding the idea of working out first thing in the morning fasted will help burn fat faster.
My best guess is this myth is formed around the idea that once you wake up and have not eaten anything since dinner your glycogen stores will be low, which means during your workout you will use fat as fuel to burn instead.
I can see why people would think this idea works since it's been 8 – 12 hours since you last ate, your body still is actively working and using calories and no food is going in to provide more energy, so in this slowed metabolic fasted state body fat stores can be used in place of the glycogen. Then why not just stay fasted through the morning?
There is an old adage that states: "fat burns in a carbohydrate flame," which is really saying that the body needs glucose (Carb Source) to lead the fat burning. With less than suitable glucose accessible to keep the body up and running correctly (like running a car on fumes), exercise intensity won’t be maximized, which in turn won’t maximize calories burned.
Ever heard of a runner hitting the “wall” during a race or hard workout? They hit the proverbial wall, not because they ran out of fat stores, but because they have a lack of glucose to keep their body running efficiently. The body doesn't just keep going until you run out of fat and that's a good thing!
So, the gist is this, if you do not eat before you exercise, you decrease your body’s capability to maximize fat burning. This also sets your metabolism up to be slower than usual for the rest of the day, which can affect any weight loss goals you may have. If you break the fast (notice where the name comes from? Break the fast, breakfast) before you go to your session, the body has the possibility to compete at a higher level, you'll burn more calories overall and your muscles will recover more effectively.
Sometimes all you need is a simple carb to kick start the metabolism before the workout. Try eating a piece of fruit like a banana or a piece of toast (sure, a little butter won't hurt). It may take some trial and error in the amount of food and timing, but once you get it right you'll feel much better during and after workouts.
Keep training smart!