enthalpy - What is meant by electrons moving to lower potential energy?

I am currently learning about chemical bonds in chemistry, and came across somebody who wrote this (source):

"Chemical bonds certainly "contain" potential energy, and the atoms want to move to a lower potential energy (become more stable).

When methane, CH 4 , forms, the valence electrons end up in more stable (lower energy) C-H bonds.

These bonds are fairly strong, so methane is relatively inert.

However, if you add energy to the methane in the form of a flame or a spark in the presence of oxygen, some of the molecules will have enough energy to overcome an activation energy barrier.

Some of the C-H bonds will break.

The electrons can then enter an even lower energy state by forming C=O and O-H bonds rather than staying as C-H and O=O bonds."

What is meant by this? What does it mean when electrons enter a lower energy state, do they move to an even lower ground state?

Answer

What is meant by electrons moving to lower potential energy?

In a chemical reaction, the electronic states in the reactants are different from the electronic states of the products because the atoms are arranged differently into molecules. Lower potential energy of the electrons is associated with stronger bonds. When the products have stronger bonds than the reactants, energy is released in the reaction (exothermic). Conversely, when the bonds of the reactants are stronger than the bonds of the products, energy is absorbed in the reaction (endothermic).

do they move to an even lower ground state?

Yes. For example, the ground state in the hydrogen molecule $\ce{H2}$ is lower than in the hydrogen atom $\ce{H.}$ (because there is more room for the electrons as they are interacting with two nuclei instead of one). If you want to break the molecule into atoms, you have to provide energy to move the atoms apart (i.e. break the bond) because separating them will result in a higher ground state. For molecules reacting to other molecules, the energy balance depends on the relative strength of the bonds (and the number of bonds).

Follow-up questions

[OP's comment]: Could you just elaborate on the part where you say "because there is more room for the electrons as they are interacting with two nuclei instead of one"? Why does the ground state get lower because of that?

This might be jumping ahead a bit too much, but electrons don't get closer and closer to the nuclei (even when that would increase interactions) because "they don't like to be pinned down" (Heisenberg's uncertainty principle). If an electron can keep the same average distance to the nuclei but occupy a large space, it will lower its energy. This is reflected in the color of molecules with conjugated double bonds (carotene in carrots, for example), and also in a model called particle in a box. One analogy is strings of different length in music, where longer strings produce a lower pitch (lower energy) and short strings produce a higher pitch (higher energy).

[OP's comment]: I read this online. It talks about how the bonds formed in the products after exothermic reactions have lower potential energy. Is it the electrons which have lower potential energy or the new molecule? This answer states that it's the electrons which have lower potential energy, which one is more correct?

Typically, adjectives for bonds are strong or weak. If you are actually measuring an energy, it is the energy of the entire molecule. The reason energies are different is because the electrons have different energies (nothing happens in the nuclei). So saying electrons have lower potential energy makes the most sense to me. Distinct bonds don't really exist anyway, we just use them for making sense of the molecular properties.

[OP's comment]: So are bond strength and energy of the entire molecule both dependent on the electron energy?

Yes, much of chemistry is about the electrons. Chemical bonds and intermolecular interactions both are a reflection of "what the electrons are doing", and chemical reactions go along with "electrons doing something new", so knowing the electronic states of the molecules carries enormous predictive power.