Let's simplify.
1) Passing electricity through metal submerged in water makes flamable gas.
2) This gas gets sucked into the intake and explodes in the cylinders, thus reducing the amount of petrol you need to run the engine.
3) Using less petrol gives you better mileage.
Do we really need a bunch of complicated formulas for this? Ok, proceed with the "over-simplifying" responses

Disclaimer: The guy in this video isn't me