This thing plugs into the wall, and yet it has an internal chargeable battery? How does that work? Do I have to plug the charger in a separate outlet?

REchargeable battery allows for the retention of time and programmed on/off settings in case of a power failure or other interruption of power.

The purpose of battery is to hold it on your hand and set up the timing function .Because the setup procedure of this product is more complicated. When setup is complete ,plug in into the wall socket . I hope my answer is helpful to you .