Aha! The supplemental materials contain all the interesting stuff, like some cool photos of the excavation site:

And section 2 goes into details about how the soil samples were collected and prepared for luminescence measurement, including keeping them in the dark/under red light lab room conditions while they were washed in acid for 3 days, before shining a laser on them. They even stuck gamma radiation probes into the holes they dug the samples from to measure the current background radiation levels there.

Interestingly in their calculation, only 5% of radiation comes from cosmic rays, and 95% from decay of nearby radioisotopes of uranium, thorium, and potassium. They were worried whether cyclical sedimentation/erosion activity of the adjacent river would change the thickness of the soil overlying the archeological site, thus affecting amount of cosmic radiation reaching it. Turns out for the final age calculation it didn’t make much of a difference whether the soil was usually 5 meters thick or 10.

I would still need to read the reference papers to figure out how accumulated radiation is related to luminescence under a laser in the first place.