New, 800V, electric cars, will recharge in half the time

Battery cells generally have a rating on their maximum charge and discharge rate which is specified as a time: 1C means the battery will charge in an hour, 0.5C means two hours and 2C means half an hour. The nice thing about this is that it then doesn't really matter how you arrange the cells into a battery (whether you put them in series for a high voltage or parallel for a high current): the maximum charge time will remain the same, assuming you don't introduce any bottlenecks. It also means that larger battery packs have a higher peak charging power. It's not very difficult with modern EV batteries for the charger connection itself to become that bottleneck: this is the kind of area where higher voltages will improve charging.

Higher voltages are also beneficial for other reasons: it reduces the amount of copper and weight you need in the car, enables smaller and cheaper motor designs (though you need more gearing as the motor RPM will also increase), and as power electronic devices improve, allows more efficient and cheaper electronics. Higher current systems do not at all scale in the same way, since with higher current there's basically no way other than just adding more material to carry it, and all that material tends to be heavy and expensive copper (or aluminium). High voltage, on the other hand, is getting easier and easier to handle as technology progresses.

The 400V vs 800V is nominal voltage for both the output from the charger and the battery pack. There's some trickery I've heard about with switching half the pack from parallel with the other half to series while charging. But generally the voltage across the output from the pack will be equal to the voltage coming from the charger.