The detector itself is a series of lead plates with interwoven layers of light-sensitive material which has then scintillator counters to detect events in the light sensitive stuff. I don't fully understand the details for the detector. (In particular I don't know how they are differentiating tau neutrinos hitting the lead plates from muon neutrinos or electron neutrinos) but I naively presume that there's some set of characteristic reactions which occur for the tau neutrinos and not the other two.

There is a conserved quantity* for elementary particles that is called "lepton number." It is defined such that leptons (electrons, muons, taus, and their respective neutrinos) have lepton number +1, and anti-leptons (positrons, antimuons, antitaus, and antineutrinos) have lepton number -1. Further, the presence of each flavor (electron, muon, tau) is conserved between the particles and the corresponding neutrinos.

For example, take the classic beta decay. A neutron decays to a proton, an electron, and an electron antineutrino. The neutron is not a lepton, so lepton number must be conserved at zero. The electron has lepton number +1 and the electron antineutrino has lepton number -1, totaling zero, and the "electron" flavor is conserved between the two of them.

Now, think about an inverse beta decay: an electron antineutrino combines with a proton to form a neutron and a positron. The electron antineutrino has lepton number -1, and so does the positron that is created; again, the "electron" flavor is conserved.

How does this apply to tau neutrinos? Reactions similar to an inverse beta decay occur when the other flavors of neutrinos interact with particles in the detector, but their flavors must be conserved, too. So, when a tau neutrino interacts, it produces a tau particle. A tau can be distinguished from an electron or muon in the detector by its mass and how it decays.

*This conservation is actually violated by neutrino oscillations, but it still holds in most other interactions.