But what is the nuts and bolts of how it is done? What do you capture the light with? How are you propelled? Are we talking solar panels or sails or some other technology?

Maybe depends on what you want. Do you want momentum transfer (thrust)? A sail, because you get more by reflecting than by absorbing. Or do you want energy (more than just kinetic)? A solar panel, because you only gain kinetic energy from a sail. If you want both energy and momentum, tune the solar panel/sail ratio to get what you want.

Lightsail. You carry a laser and a lightsail. You shoot the laser at the black hole and the laser light gravitationally slingshots around the black hole, returning to you with more energy than you gave the laser. The returning laser light pushes your lightsail. The black hole needs to be moving for this to work, although it doesn't really matter what direction the black hole is moving.

It does very much matter which direction the black hole is moving; it needs to be moving toward you or you won't get any blueshifting effect, which is the entire reason the energy budget works out.

The idea in the paper is to put up a big solar panel. Simply by absorbing the light, you are propelled. And you use the energy in the light, which is more than the energy you used to send it, to send more light.

I think you might gain momentum simply from absorbing the light.

You gain even more momentum from reflecting the light back into the black hole! When you absorb a photon, you gain its momentum. However, when you reflect a photon, you gain twice its momentum to account for the momentum possessed by the photon now flying away from you.

Isn't that a three-fer?

You first emit the photon towards the black hole, momentum gain. Absorbing the photon on the return would be a two-fer, and reflecting a three-fer.

Absorbing the photon would be a two-fer. Reflecting it is just starting the next two-fer. It's two-fers all the way down.

Ooh. Even better, it's like compound interest.

You emit photon 1. It has momentum m. It comes back blueshifted, so you get momentum km, with k > 1. You reflect that back, and you get momentum k^2m...

It's not that simple, because as you start to move away, you redshift the photons you reflect. And the alignment issues are going to be pretty severe. And if you get too much compound interest, the photons may be blue-shifted enough to kill you or damage your ship. Still, the idea is pretty slick.

Also in all slingshot maneuvers you get more the closer you are, so you wanna pour in the most energy the closest you can get.