Yes, the helium is necessary. It dilutes the CO2 in your exhaled breath, so that your CO2 levels stay low while your O2 levels drop, thus avoiding the activation of the breathing reflex (since the breathing reflex is activated by high CO2 levels rather than low O2 levels). Pure nitrogen would work too, but helium is probably easier to get hold of. With just a plastic bag and ordinary air, you'll quickly start fighting for breath and become extremely uncomfortable, just like you would if you held your breath.
