To start with, they do a lot of work to eliminate radioactive backgrounds in the materials they're using, and they put the detector deep underground to shield from cosmic rays.
Additionally, when a particle interacts with the nucleus, the ratio of how much energy ends up as scintillation light versus ionization is different than when a particle interacts with an electron, which is most of the background processes.
Then, whatever is left, they try to model using known processes. After all that, there's one event that they can't account for. And that's what the news is about.
I agree. Moreover, I'm not sure if it's the same team, but in a similar experiment while removing all the other effects, they discovered that Xenon 124 is radioactive, but the half life is super long and no one had seen it before. https://xenonexperiment.org/observing-the-rarest-decay-proce...
Yeah, it's funny, for experiments like this you spend 90% of your time modeling and subtracting noise, and 10% analyzing the signal that results. Had the same experience in X-ray astronomy. 3 years building a detailed model of all the sources of noise, then subtracting it out and finally starting on the science.