A watt is a derived unit for a rate of change, an amount of energy used in a unit of time, so P = E / t. A kW per hour would be a rate divided by time, or E / t^2, resulting in another rate.
More colloquially, think of watts/power by analogy to another rate, that of speed. Moving at a speed of 100kph for 3 hours results in 300 speed-hours of distance. Saying 100 kilometers per hour per 3 hours sounds awkward, but is actually a weird way to say acceleration, a rate of change of speed. (And probably a hint to get your car serviced.)
Anyway, the key is to think of a kilowatt as a rate, not a quantity.