Does it raise your BMR that much? What is that energy doing exactly? Does your body temperature increase? Do you move around more? You can't have energy going nowhere, of course. What other activities can drive the same? And what is the magnitude?
"The extra muscle mass you built also raises your BMR in a longer lasting way."
Unless you are talking about fantastic amounts of muscle mass here -- 10 pounds of muscle will only burn an extra 60-70 calories a day -- this oft mentioned factoid is unnoticeable in daily calorie expenditure.
It's hard to find good data on what the effects are in amongst all the bro-science articles, but I believe the raise in energy usage is due to the body repairing the muscles after the weights workout, so you'd see an increase in body heat generation (although the core temperature would stay the same, peripheral circulation would go up and skin temperature would rise a bit). HIIT is just good at hitting a lot of different muscle groups hard enough in a short amount of time to maximize the effect. It's not huge but several hours of increased passive energy burn does make a difference.
Also, 10 pounds of muscle isn't a fantastic amount to put on (depending on body size), especially if it's muscle that you're re-gaining rather than building for the first time. A few months of regular moderate weights work would build that much without pushing too hard. And again, passive energy burn is "free" and helps with gradual long-term weight loss.
"The extra muscle mass you built also raises your BMR in a longer lasting way."
Unless you are talking about fantastic amounts of muscle mass here -- 10 pounds of muscle will only burn an extra 60-70 calories a day -- this oft mentioned factoid is unnoticeable in daily calorie expenditure.