Good analogy for watts vs amps?
The water analogy is full of leaks. A better analogy, if you need one, is a closed circuit water pumping system such as a central heating hot-water system. Pump = battery, pressure = voltage, radiators = resistors, current = current.

Forget all that for a moment: why does voltage affect power? Increasing the voltage allows us to get the same current through a higher resistance. Since more work is required to do this then the power must be increasing.