The mental clarity comes from adrenaline. When you fast your body slowly uses up its glycogen store and releases adrenaline to compensate. Presumably it does so in order to keep you active / motivated enough to go look for food to refill your glycogen stores. It is this adrenaline that gives you the mental clarity and causes you to have trouble sleeping when on a fast: fasting is essentially natural adderall.

On top of this, once you use up all your glycogen, your body enters a period of autophagy which causes disfunctioning cells and protein fragments to become broken down and reused. This causes your body to function more efficiently, including your brain.