Blazars are a type of active galactic nuclei (AGNs), which are themselves galaxies with a supermassive black hole at the center. If the black hole is surrounded by a dust cloud (or other matter), it will gradually accrete that matter. In the process, it will eject a fraction of it in a relativistic jet perpendicular to the galaxies axis of rotation. The jet is turbulent, and thought to be a likely site to accelerate particles to extremely high energies. In blazars, this jet is pointed nearly directly at Earth, giving us the best chance to see these ultra-energetic particles.
The story begins on Sept. 22, 2017, when IceCube observed a neutrino with an energy around 300 TeV (about 50 times the energy of the protons accelerated at CERNs Large Hadron Collider). The event display shows the neutrino; each colored dot shows the photons registered by one IceCube optical module, with color indicating relative time (from red to blue), and the size indicating the number of photons.
IceCube has seen many neutrinos that were more energetic than this, but this was still enough energy that the neutrino was likely of astrophysical origin. So, computers clacked and whirred, and 43 seconds later we sent out an automatic alert, telling many partner observatories that we had seen an energetic neutrino coming from a specific direction. Several of these observatories pointed telescopes in the direction of the neutrino, and (to make a long story short) that the location coincided with a minutes to years; this neutrino came during a time when TXS0506+56 was emitting at particularly high levels, from radio waves through (at least) 1 TeV photons. The first paper, by IceCube and the other experimental collaborations, discusses this coincidence in space and time. TXS0506+56 is a relatively energetic, quite nearby (with a redshift of 0.3365), so it is a likely candidate for a first observation.
Shortly after this observation, IceCube went back and looked at archival data, searching for excess emission from the source. We found an excess of neutrino events coming from that direction during the period from September 2014 to March 2015. This is reported in the second paper.
The exact statistical significance of these observations depends on some of the details of the analysis - the first paper gives a range of significances, depending on the preferred assumptions. But, taken together, this is strong evidence that we have seen neutrinos coming from a specific source: we have found at least one cosmic accelerator, far more powerful than CERN's LHC. Besides the observed neutrinos, there is strong suspicion that AGNs also accelerate the ultra-energetic protons and/or heavier nuclei cosmic-rays that led us to look for neutrinos in the first place. Unfortunately, since protons and heavier nuclei are bent by interstellar magnetic fields, they do not point back to their sources.
One still-open question is whether blazars are responsible for all of the neutrinos that IceCube sees. In 2016, IceCube published a paper (freely available arXiv version here) which set limits on the fraction of the astrophysical neutrino that could come from blazars, setting a limit between 27% and 50%, depending on the spectral index. This paper studied 862 blazars, and had to make some assumptions about the relationship between the observed gamma-ray flux and the expected neutrino flux. As you can imagine, extensive work is ongoing to revisit this question.