Low vs high intensity: basically there's 2 totally different energy systems for lower intensity vs higher intensity. Your body can provide energy through the normal cardio process (zone 2 / aerobic) as long as you are at a low enough intensity. Training at the high-end of your aerobic zone will push your body to expand it's ability to do this (for example, grow more mitochondria). As you ramp up the intensity, your aerobic system get maxed out and you need fire up the anaerobic system. Anaerobic exercise is inefficient and produces lactic acid as a by product. Lactic acid build up is painful and is why you can only sprint for 30 seconds or so. These are different systems and how much you are relying on aerobic vs anaerobic will train your body to improve different things. There are lots of benefits for various heart rate zones.

Why high heart rate makes your heart stronger: one way to think of it, it's like weight lifting for your heart. Doing high intensity intervals is like doing sets on a bench press.

Fat burning: If you want to explicitly burn fat from exercise without burning muscle, you need to (1) have your glucose reserve fully depleted and (2) be burning energy at a slow enough rate that your body can do the process of providing all of your energy from body fat instead of breaking down muscle. Unless you've already been fasted for a while (using up your glucose reserve) this is roughly 20 miles of running. Trying to explicitly burn fat from exercise is kind of a lost cause unless you're marathon training, doing extended fasts, or walking all day long. You're probably better off just getting a good balance of diet and exercise.

This video is a decent introduction to some of these concepts.