It would be wonderful if it had been a simple race between light and neutrinos, and the neutrinos arriving first. Sadly, it is not as simple as this.
In reality, "trains" of millions and millions of neutrinos are generated in Switzerland, from where they travel straight through the earth, to be detected in an experiment in Italy. Over three years of measurements, 16111 neutrinos have been recorded and timed.
Each train of neutrinos is 10500 nanoseconds long, and the average neutrino is measured to appear 60 nanoseconds earlier than expected. The problem is that of the millions of neutrinos sent, only a few get detected.
So when the arrival of a neutrino is measured, it is not certain if it is one from the beginning of the 10500 nanosecond long train, or one at the end.
If the train of neutrinos were exactly uniform, you could just take the average travel time, and be done with it. In the end the length of the train would average out. The problem is that the density of neutrinos being sent is absolutely not uniform.
Through crafty statistics however, and by measuring the precise density shape of the train in Switzerland, it is however possible to still generate an average travel time. And this is the number that is being reported, between 50 and 70 nanoseconds too fast (more or less).
This corresponds to around
It is an impressive effort. It would have been a lot easier if we could just organize a race between a neutrino and a photon. The result now reported is statistical in nature, but the statistics are impressive. It should however be realized that a lot of calculations are needed to get the 60ns number, and a mistake could hide anywhere.
The result can be compared to trying to measure if a (real) train runs on schedule by timing when people walk out of the train station. If you keep that up long enough, you will be able to get results - but a lot of things could mess with your measurements!
On a final note, it might well turn out that neutrinos do not travel faster than light, but that there is another reason why we are getting these results. That might in itself be almost as interesting!