Simply and clearly, how does the energy get inside a battery?

The battery makes it. The elements that are assembled to make the battery produce the electricity as they react to each other. The reaction doesn’t start until the posts are connected by something that will allow the negative electrons to flow out to the positively charged side of the battery.

@WestRiverrat Thanks!

The battery transforms it.

This video is good.

Show your daughter an example of chemical energy transforming into kinetic. Drop some vinegar into a small pile of baking soda and watch the volcano erupt.

Inside the battery are two pieces called electrodes. These electrodes are attached to the terminals you can see on the outside of the battery. One is positive, one negative, just as labelled.
The electrodes are surrounded by a material called electrolyte. The electrolyte is a material through which electrical charge can pass, but only in a certain way.
Through interactions with the electrolyte, the electrodes can change their state of charge relative to one another. This means that one of the electrodes has more electrons attached to it (sort of, not really, but it’ll do, it’ll do…) than the other has. One is “full” while the other is “starved”.
The potential energy in a battery results from this difference in charge.
The electrical energy from a battery results from electrons flowing from a “full” electrode to a “starved” one.
The energy gets into the battery in one of two ways: either the materials of which the battery is made have a charge difference to begin with when the battery is made, or an electric current is applied to the terminals to force a charge difference between the electrodes.

The science that explains how batteries work is called electrochemistry if you want more detail.