This is the best attempt to date to directly measure the mass of the electron (anti)neutrino. When a nucleus beta decays, it emits an electron and an antineutrino. If the neutrino were massless, the electron could carry away the entire energy of the decay. What they've tried to observe here is the highest energy electrons, to see if they can see the effect of some of that energy going to the neutrino mass. It's an amazingly difficult measurement, since only a very small fraction of those electrons have the highest energies.
Measuring the neutrino mass this way would be relatively unambiguous, which is why it's exciting to see progress.
That said, there have been other experiments that have put stronger constraints on neutrino masses, but only for certain models. Double beta decay experiments look for decays in which the two neutrinos annihilate each other, allowing the full energy to be emitted by the electrons. However, this requires the neutrino to be its own antiparticle. This is allowed, because neutrinos are neutrally charged, and wed call such a particle a Majorana particle. The rate of these neutrinoless double beta decays would tell about the neutrino masses, but only if neutrinos are Majorana particles. The exact measurement would also depend on how well we understand the energy levels of the nuclei involved. So far, the upper limits for neutrino masses from these experiments are on the order of 0.1 eV.
And there are cosmological constraints on the neutrino mass from experiments that look at the cosmic microwave background. Early in the universe's history (like the first second), the mass of neutrinos would have influenced how much matter clumped up due to gravity, which would lead to fluctuations in the microwave background. Modeling this requires us to understand particle physics very well at those early high energies, so there's some uncertainty due to modeling. But again, the limits from these observations are on the order of 0.1 eV.
So we now have one direct observation that's consistent with other, model-dependent observations. The most interesting scenario would be that KATRIN actually observes neutrino mass as it gets more data, implying that our models might be wrong. But even if the experiment doesn't, it's still great to have the extra constraints.