Long slow steady runs (at least 90 minutes worth of running at conversational pace) is what will train your fat burning engines. There is also some studies about running your LSD fasted that is supposed to help train your body to do this too. Google "train low race high".

The point of training at lower heart rates (or slower paces) is that you are running and using energy where you are supplying enough oxygen to the muscle cells that need them to burn energy aerobically. You can only burn fat for energy when your muscle cells has enough oxygen. The faster you run, the more you force your body to burn fuel anaerobically, which means you are depending more on glycogen stores and glucose and less on fat.

If you run too fast, yes you are running anaerobically. The blood can deliver only so much oxygen to your cells. Your muscle cells can only absorb so much of this oxygen from the blood to burn fuel (look up running economy). If you run too hard, you are creating a demand for energy that your aerobic system cannot keep up with and you go beyond your aerobic threshold. You are also now producing more lactate and hydrogen ions which makes your blood acid levels rise which limits how far you can run (running beyond your lactate clearing threshold). Running slower (at conversational pace) allows you to run farther and longer.

When you run at these slower speeds (70-80% of max heart rate) you will improve running economy by forcing the body to produce more hemoglobin (its what carries the oxygen from your lungs to muscle cells in your blood) and increases the amount of mitochondria in your muscle cells (its the oxygen burning part of your muscle cell). You can also make your heart stronger which delivers more blood per tick to your cells. All of this happens when you run at 70-80% of your max HR or conversational pace.