The paper suggests the mechanism towards the end. I'm not a physicist, but here's my summary (I'm sure full of errors, but I think this is roughly what they're proposing):
The paper attributes development of this engine to developments along the de Broglie-Bohm pilot wave interpretation of Quantum Mechanics as opposed to the Copenhagen interpretation of Bohr and Heisenberg, which is more generally accepted. (https://en.wikipedia.org/wiki/De_Broglie%E2%80%93Bohm_theory. Einstein, for instance, never accepted Copenhagen.. famously saying God doesn't play dice). Their approach is called the "pilot-wave" hypothesis (nice simulation here https://youtu.be/nmC0ygr08tE), where a wave and a particle are both epiphenomenon of certain resonant frequencies (Chladni patterns/Faraday waves) in a more complex base wave (as in the video). This base wave is a normal acoustic wave in the "medium", but it wasn't understood for a long time what that medium could be. After all, the "ether" of the 1700-1800s had been rejected. Now, they're saying the medium is quantum foam, which is some electron-positron bubbling that it seems just permeates space (also responsible for zero-point energy/casimir effect), and that's what is responding to the photons in the chamber. The chamber reflects microwaves (photons of a certain frequency) back and forth, and in one direction they "push" harder than the other, due to expansion and contraction controlled by the shape of the chamber. By analogy, it's like paddling in water; you put the paddle in the water in one direction, and move it back through the air in the other; in both directions you're moving through a medium, but you control it so that you impart more momentum in one than the other. Except instead of paddle in water and air, here the momentum transfer is from the microwave photons to the electron-positron foam permeating space. That's why no propellent is needed.. it's pushing against something that's really there, in the chamber and all around. So, space isn't empty and we can row our space boats through it.