I think the Oberth effect is easier to understand without referring to conservation of energy or momentum. I don't know the etiquette of repeating answers here but if anyone objects I'll try to figure out how to post a link to another question where I gave essentially the same answer, and remove this one:
The Oberth effect can be explained this way. Every second you are falling in a gravitational field, it is altering your velocity--perhaps helping you, perhaps hurting you. If you do your burn when you're deeper in the gravity well, you're either increasing the time gravity will help you, or decreasing the time it will hurt you. For example:
- If you're moving away and you want to slow down, burn now. By slowing
down, you spend more time near the planet with gravity pulling you
- If you're moving toward the planet and you want to speed up, wait
until you're closer. If you burn now, you spend less time falling
toward the planet getting accelerated by gravity.
...and so on for the other cases.
The equations and laws tell you that the Oberth effect will happen. They don't tell how it works. If your capsule is going faster, something had to push or pull on it. In this case, that thing was gravity. Conservation of energy doesn't speed things up or slow them down, it just tells you that something will do so.
(Related: is there a tag for soapbox? Tired rant, maybe?)