The purpose of this electrolysis device that splits water is to increase the fuel efficiency of the automobile, as also suggested by mrjeffy321. The electronic control module (ECM) controls many functions of how the car engine operates and makes modifications accordingly. In this case, the most important one is the amount of fuel that is injected into the cylinders. With the addition of hydrogen and oxygen, the combustion of gasoline is more efficient, so the ECM "tells" its attached components to inject less fuel into the cylinders because the car's energy needs are satisfied.
This is not entirely correct. For cars with ECMs, a MAP enhancer must be installed in order for the system to work (work as reported, that is).
When brown's gas is added to the air stream, additional O2 gas is picked up by the O2 sensors in the car's exhaust stream. The ECM responds by ADDING more fuel to the system. This actually decreases the car's efficiency. This is where the MAP enhancer comes in. The MAP enhancer is situated between the ECM and the oxygen sensors. They are usually adjustable and modify the signals (changes the voltages) from the O2 sensors such that the ECM thinks there is less O2 in the exhaust stream than there actually is. The ECM then responds by reducing the amount of fuel delivered to the engine in an effort to preserve, what it thinks is, the optimum fuel/air ratio.
The ECM is actually "blind" to the actual effiency of the car; it's just responding to false signals given to it by the MAP enhancer.
This is not quite accurate. Oxygen sensors respond only to excess oxygen in the exhaust stream, no matter where that oxygen comes from. In general, auto ICEs, being presumed to have fairly complete combustion, should have nearly zero excess oxygen.
Brown's gas is a mixture of hydrogen and oxygen from the hydrolysis of water and has no excess; burning the hydrogen requires all the oxygen produced by the hydolysis.
The so-called controls put in series with the sensor merely bias the sensor signal so that the engine then runs lean. A lean engine often gets better mileage (you can do the same thing by just using a smaller engine), though the effect is not terribly large and has some undesirable side-effects.
Altering the signal thusly is not good for the engine and is illegal in many locales. At the least, you cannot pass an emission inspection with that bias in place.
I have built many multi-gas burners (including hydrogen) and have monitored them with this kind of oxygen sensor (generically called a Nernst cell) and they required no change in the bias to adjust for the type of fuel, provided that reasonably complete combustion took place. Obviously such things as coke ovens put out a lot of CO that burns poorly and doesn't always produce the optimal results at standard settings; this is not true of gasoline.