How do electrons carry energy
'How' is a question that's impossible to answer.

You have to start with the observation that when you connect a battery to a resistor, energy is transferred.

Now there are several levels of model we can invoke to summarise how much is transferred. We can also push those models to 'explain' what's going on in other terms that we feel good about, that we think we understand.

The simplest model is circuit theory. The battery delivers a current, from a voltage, and the product is power. It doesn't tell us what voltage is, or current consists of, but it's neat for doing calculations ranging from the very simple to the very complicated. The hydraulic analogy is often invoked to make it more familiar.

Next we could think about mobile electrons. Voltage is a measure of how much energy per unit charge is involved when they move. It doesn't tell us what electrons are, or how energy is transferred. There are a lot of electrons in the conductor, some move, most don't, why the difference? They gain energy by moving 'down' the field from the battery. How does that get transferred to the resistor? Is there friction between the electrons and the atoms in the resistor? Do they hit the atoms? Do they get scattered? All three are defensible explanations, depending on the level of model, and what you want to assume.

At the lowest level of our understanding is quantum mechanics, where electrical effects are mediated by photons (yes, even at DC). Frankly, that does my head in, because I lack the imagination and skill to work with those models. But it underpins the simpler models, and it still doesn't tell you what's 'really' going on.

Just get used to the idea that energy transfer happens, and learn enough equations at the simplest possible level of model to allow you to figure out how much.