Dimensional analysis & how go get between things.
The ‘power company’ is really the energy company. They supply energy. The rate at which you consume their energy is your power draw, measured in Watts. The amount of energy you pull from them is what you’re billed for. That’s measured in kilowatt hours (kWh).
Batteries are measured in Amp-hours (or milliamp-hours) because they put out a certain Voltage. They put out that Voltage at however many Amps for an hour. That’s [Volts]x[Amps]x[Hours], which is the same thing as [Watts]x[Hours], which is the same thing as energy.
Source
WolframAlpha