I think the original quote was about what to eat before exercise, not cutting out carbs altogether. I'm sceptical too, aren''t carbs the body's preferred and first form of fuel? But I have been told that you can sort of wean yourself over to using more protein and fat instead, but I think this takes some getting used to.

But this one I'm pretty certain of: "fat-burning workouts" are largely a misunderstanding. As I recall, studies showed that low- and medium-intensity workouts burn more fat than high-intensity workouts during the workout. But in total, considering the muscle you build, and the after-effects of the exercise, high-intensity workouts were just as useful for burning fat, it was the total effort (calories out) that counted.