Oh, they finally bothered to add slides to the conference website.
I think the BBC introduction does quite a good job. We expect that initially there was the same amount of matter and antimatter around. If they come together they annihilate. If matter and antimatter would be completely symmetric, nothing is left. But there is matter left - there has to be an asymmetry (called "CP violation" - details don't matter here), and we can calculate its strength based on the amount of matter that was left.
CP violation has been observed, first in 1964. The problem: the observed violation was tiny. We found it in a few more related places since 1964, but always too small.
The hope is now that neutrinos show larger CP violation. Unfortunately, neutrinos rarely interact with matter, so all those measurements are complicated, need large detectors and a lot of time. The result of NOvA has a weak hint of non-zero CP violation for neutrinos, but the measurment uncertainty is still very large.
@Orodruin is a neutrino expert.