To shed a different light on this: think of temperature as walking through mud. Your legs lose energy trying to slowly pull a lot of mud behind you. Now think about skiing. A lot less snow is dragged with you but it flies fast.
Here, what is interesting is if one fusion reaction does happen, then the alpha (helium) particles leave at 2.9 MeV. After two collisions with protons, if the second proton they have hit hits in turn a boron nucleus, then it will have exactly the right energy (612 keV) to have maximum chances at initiating a second fusion reaction.
612 keV is like almost 7 billion degrees °C if considered as thermal energy, and no experiment anywhere will get that hot for long. But compared to the energy of the exiting helium nuclei, it's still much lower (0.612 MeV vs 2.9 MeV).
In other words, instead of cascading all the energy down and hoping the sea of particles rises to a few billion degrees so enough particles do fusion to keep the sea of other particles hot, here, the energy is preempted by proton atoms after just 2 collisions and used immediately to start a second reaction, which yields more helium nuclei at 2.9 MeV, essentially producing an "avalanche" effect.
Finally, yes, they seem to have devised a way to obtain at least a small part of the energy electrically, without relying on thermal energy, via direct electric field deceleration of very fast charged particles.
This is like "the ultra rich (very fast particles) manage to create value among themselves without having to cascade their wealth down to the crowd (cold particles), and then upload that value to hyperspace (the electric field from the electrodes), without ever interacting with the mass of the crowd (the mass of the target), until a sufficient amount of fusion reactions have been realized"
The avalanche process is explained in Hora's 2016 publication, with a schematic page 9: https://aip.scitation.org/doi/10.1016/j.mre.2017.05.001
And yes, a petawatt (the energy of present day ultra-fast lasers) is a lot of power. It was just chance that there was very little practical use to this kind of power - until now.
That being said, I am not a true expert myself of this topic, so the true barriers laying in front of this concept might be better explained by the other comments here.