I'm sure you don't actually mean this, but the order in which you've written your information at least IMPLIES that low insulin has something to do with more fat being burned. And I'm almost positive that has nothing to do with it.
A low carb diet generally means lower glycogen stores, and the lower your glycogen stores, the greater proportion of fats are burned for energy. The higher the glycogen stores, the more of the energy required for physical activity will come from glycogen.
Yes, I'm aware I wrote "low insulin" and then "low carb" in the next paragraph. That's what I mean: you started talking about insulin and then jump to low carb (nothing wrong with the jump) but the flow of information implies the association I'm talking about.
Of course, it depends on what type activity you're doing, but again I think you forgot to mention that more fat being burned on a low carb diet is due to lower average glycogen stores, not less insulin spikes. At least, that's what your text implied (or what I felt it implied).