The way I understand it is that you can consider the whole system of rocket and expelled propellent as a system with a fixed amount of energy, which you are allowed to change (by converting from chemical to kinetic energy, say).

However, by changing the speed you're moving at when you fire the engine, you can change the proportion of energy that ends up in the exhaust and the proportion that ends up in the rocket, while keeping the totals consistent. You want to increase the latter, and since the former is not useful, you don't mind "stealing" it.

The space of possibilities begins at zero, when the rocket is fixed to something infinitely massive like a planet (well, close enough). No matter how many thousands of tonnes of fuel you burn, the rocket will never move: all the kinetic energy is in the exhaust.

Now, consider the case of a rocket in space, at rest. The rocket exhaust carries away a percentage of the energy, but some of it ends in the rocket. How much depends on the engine, but it's not zero, and it's not 100%.

So now we start to move up the other side of the space: when the rocket is already moving, the exhaust has less energy because it's moving the other way. But this is a conserved total and the rest must be in the rocket, where it's useful. You can also "explain" it in that because the rocket is moving, the (constant) force of the engine acts over a longer distance and this imparts more energy (by W=Fd). The "cost" to this is slower exhaust, but that's not a problem: we don't care what the exhaust is doing. Another way to think about is that "fast fuel" also imparts some of it's current kinetic energy (we'll come back to this) in addition to its chemical energy.

So, we're trading something we don't care about, exhaust, for something we do: the rocket's speed.

The problem is that when you fire an engine on a moving rocket, the propellent you used was accelerated to the current speed by the rocket itself: you do get more from a kilogram of fuel when you're going fast, but you paid dearly with other fuel to bring that kilogram with you to that speed. The extra kinetic energy you are getting isn't free: it was already added to the fuel by earlier conversion of other fuel to kinetic energy of the fuel you're burning now. So "easy, just go faster" isn't useful advice: it's true, but it's like telling a person without money that they can become rich by making bigger, more efficient, investments.

The clever bit of an Oberth Manoeuvre is that you use an external effect to accelerate you to a speed where your fuel is more effective. Then, you don't need to pay for accelerating that fuel, but you get the payoff of using it when it's most effective. If you use a gravity will to provide this higher speed, you decelerate on the way out, but you get to keep the kinetic energy you gained by the engine firing, and you get to keep it at the efficiency you burned it at.

Perhaps you could make a strained analogy. Say you want to buy as many hours of in-person accounting services as you can with your money: 100,000 roubles. If you buy them at home in Moscow, they cost 10,000 roubles per hour, so you can buy ten hours.

Now, the service costs 5,000 roubles in another city because the general costs there are lower, but an hour of service is as valuable to you. You can't do the services remotely. So what you need to do is to travel to that city, buy the services and go back. Flights cost a few thousand roubles.

The Oberth Manoeuvre is analogous to that journey: it is a way to place yourself in a market where money goes further. The cost of the flight is the cost to set up an Oberth Manoeuvre.

If you have to burn more fuel to get to an Oberth Manoeuvre (maybe you're not in a good position in space to hit a good trajectory), then it's analogous to the flight costing more money: it could still be advantageous, but eventually it might not be.

The analogy to a deep solar Oberth is that there's another city that's even cheaper, but it's in a warzone (wonder why...). You can get what you want at a very effective price, but you have to also be prepared for a harsh environment while you're getting them. For example, you may have to pay for body armour (i.e. the heat shield). There's also a chance that you don't come back at all (i.e. the probe fails while deep in the solar gravity well). Your body might be sent back on the return flight (the dead probe will also climb out of the well), but that might not be much consolation.