In short, in Orbital mechanics, burning your thrusters deeper within a gravity well, results in a greater increase in kinetic energy than burning them further out.
This is because momentum ~ v, while kinetic energy ~ v^2. If you're travelling faster—as you would be as you approach the black hole and fall deeper and deeper in your orbit—then you can expend to same amount of momentum to receive a disproportionally larger increase in kinetic energy.
Because your potential energy falls off with distance to the black hole at the same rate regardless of the speed you're travelling at, your total energy upon escaping the black hole is much larger than it would be had you burned your thrusters outside its gravity well.
The phenomenon described is not due to the Oberth effect.
What would be happening would be that the human in the spacecraft, and the spacecraft, are accelerating at exactly the same rate because they are accelerating due to gravity. Thus, the human feels no pressure accelerating him (no outside force acting upon him) from e.g. his seat. And his internal organs feel no pressure accelerating them from each other. They (the craft and the human and all his internal organs) are in free fall together and feel no forces acting upon them despite the whole system (craft-human-organs) being accelerated to tremendous velocities.
It's super un-intuitive, but you're indirectly harnessing the gravitational potential energy of lowering your propellant into a gravity well and leaving it there. The overall orbital energy gain of the spacecraft can exceed the chemical energy of the fuel.
In terms of the energies involved, the Oberth effect is more effective at higher speeds because at high speed the propellant has significant kinetic energy in addition to its chemical potential energy.[2]: 204 At higher speed the vehicle is able to employ the greater change (reduction) in kinetic energy of the propellant (as it is exhausted backward and hence at reduced speed and hence reduced kinetic energy) to generate a greater increase in kinetic energy of the vehicle.
In short, in Orbital mechanics, burning your thrusters deeper within a gravity well, results in a greater increase in kinetic energy than burning them further out.
This is because momentum ~ v, while kinetic energy ~ v^2. If you're travelling faster—as you would be as you approach the black hole and fall deeper and deeper in your orbit—then you can expend to same amount of momentum to receive a disproportionally larger increase in kinetic energy.
Because your potential energy falls off with distance to the black hole at the same rate regardless of the speed you're travelling at, your total energy upon escaping the black hole is much larger than it would be had you burned your thrusters outside its gravity well.