To make a long story short: you will start burning muscle when you run out of carbs and dietary protein. Also, your body will only keep as much protein in the muscles as it has to: if you're not training in a way that requires a muscle to maintain it's size, the body will leech out some of its protein and use it for something else. If you want more details, here's the long story.
Background. To understand why protein is drawn from the muscles to be used as energy, you have to understand a little bit about how energy is made in the body (metabolism). In order for fat or carbs (or protein, as a last resort) to be fully converted into energy (i.e. burned), they have to go through a process called the citric acid cycle, which releases energy through a number of reactions. This is the predominant energy system at rest and during prolonged aerobic exercise, and the primary source of energy in these states is usually fat. But here's the thing: one of the intermediary products of the citric acid cycle (oxaloacetate) requires a derivative from either carbs or protein. Because of this, to metabolize fat, you have to break down at least some carbs or protein. A common summary of this is "at burns in a flame of carbohydrates." Your body prefers to burn carbs over protein, but it has limited carb stores. If it runs out of those, it will start breaking down protein (into ketone bodies) to be used as fuel. This is also important for fueling your brain once you run out of carbs, since it can only use carbs or ketone bodies as fuel (not fat). It kind of makes sense that your body would evolve the ability to use protein for energy as a back-up energy source considering there's so much of it in your body.
Aerobic Exercise. Prolonged cardio can deplete your carb stores (colloquially known as "hitting the wall"). For example, if you do a marathon without carb-loading in the days before the event and without consuming carbs during the race, you will run out of carbs in your body and start burning protein. According to the "hitting the wall" article on Wikipedia, carb stores can come depleted in under two hours of moderate exercise. The harder you work out, the faster you'll deplete your stores, because as you work out harder, you begin to burn a larger proportion of your energy as carbs instead of fat.
It makes sense that you would burn protein that you recently ate first, but even if that's all you burn, now you don't have that protein around to help you build muscle (not to mention all the other stuff in your body). Also, if you don't have an immediate use for protein, your body converts it irreversibly into fat (i.e. your body can't convert fat back into protein), so depending on when you ate, you may not have as much protein available as you think.
Food > Fat > Muscle. This is hard to discuss in detail, because it's an oversimplification, but I can at least comment on the spirit of it. During starvation (or dieting) muscle is preserved as much as possible, but based on the info above you can see how this hierarchy wouldn't always apply; there are some things protein can be used for that fat can't be.
High Intensity Training. I'm not very familiar with weight training HIT, but I get the author's logic. As I understand it, HIT is when you do one high-rep set to exhaustion (per muscle group). However, according to the author, doing multiple sets is more beneficial for muscle growth. I assume this is because HIT doesn't overload the muscles in the right way (there are a few different ways to overload the muscles that all correspond to different goals. Check out this ACSM position stand if you want to know more). Your body doesn't waste resources making a muscle any bigger than it has to be. If you don't constantly require your muscle to be big/strong (by doing the right type of training), your body will take unneeded protein from the muscle and use it for something else.