Imagine a bath full of water. The volume of water is like the amount of stored energy (the kWh). Now imagine a tap is opened and the water flows out. The speed that it flows out is like the power (kW). The water (energy) will pour out of a big tap, but only trickle out of a tiny tap.
For your home system, you want power sufficient to run real world equipment (a big enough 'tap'). For example, a kettle typically uses 2 kW for a few minutes. If your battery can deliver a power output at 2kW or above then it will all come from your cheap stored energy (the PowerBanx X range can output up to 3 kW).