Some scientists ended up trying to pin down more accurate decay rates for radioactive materials, and found out that they have regular variation - mostly seeming related to the sun! This is extremely strange, as the article says - but in ways that the article doesn't even mention. (via Flatrock.)
They talk about a dip in decay rates a day and a half before a solar flare happened. Then they talk about neutrinos being the best current candidate for causing the change in decay rates. Neutrinos are produced in huge quantities by fusion in the core of the sun. They travel at (or just below) the speed of light and go straight through everything, which is why it's a bit puzzling how they could affect radioactive decay.
The current model of how the solar interior works says it takes thousands of years for anything other than neutrinos to reach the surface from the core. If neutrinos are the source of the dip in decay, they have to also be the cause of that solar flare - because there's no way the flare produced the neutrinos that caused the change in decay rate. At best, the mechanism that started the flare could have also altered the neutrinos; I'll get into that later.
Similarly, they talk about the 33-day cycle in decay rate changes and speculate about the rate of rotation of the sun's core - but neutrinos radiate straight out from their source in all directions regardless of whether the sun's core rotated in 33 days or 33 minutes. Current theory would say this is due to some 33-day cycle of changing intensity of fusion, not rotation.
If the rotation and solar flare points presented in the article are correct, it means there has to be some other effect happening than just quantity of neutrinos. I hope they aren't just misstatements due to the writer oversimplifying the mechanisms behind neutrino production and propagation.
The correlation between the solar flare and dip in decay rates is very interesting, though - whether because the same solar event caused both, or the flare caused the dip, it's exciting new science. Of course, it could also be a false relationship; they could have happened for unrelated reasons. Proving that they're related will take a lot more testing, and under unpredictable circumstances - every flare is different, some might be associated with a decay dip and others wouldn't, and if it IS affecting the neutrinos, it would have to be aimed right at Earth for us to detect it.
Now to go even further into the physics...
Neutrinos are strange particles and difficult to study. With large particle detectors buried far enough underground to block out everything else, we can detect a tiny fraction of the ones passing through the earth every second. Even better, there are three kinds (called electron, muon, and tau) and only the electron neutrinos are easily detectable even with those means. Most of the ones produced by the sun are that kind, but during the trip through the sun and space they convert to the other kinds; we only see about a third as many electron neutrinos as theory says are being produced, and they're all constantly converting back and forth between the three types. This can be modified by what they're passing though, to change more of them to a specific type under one set of conditions vs. another, but overall it's a pretty chaotic process with a well-tested result that a third of them are electron neutrinos by the time they get to Earth.
However, it's possible that one of the other types is what's affecting the decay rates. If so, rotation of the sun could have an effect, because the slight variations in conversion rates to different types could produce an excess of muon neutrinos reaching us from one part of the Sun's surface and an excess of tau ones from another. If the type of neutrinos passing through radioactive atoms affects the decay rate, this would matter. Perhaps that flare produced a concentrated effect of suppressing conversion to tau neutrinos, causing the output to be 33.3% electron, 33.7% muon, and 33% tau - more than enough variation to cause the dip in decay rates seen by the experiment. This is interesting if accurate, because we'd have changed radioactive decay from "random event over time" to "every time a nucleus is hit by a tau neutrino this much energy is added to it, and if that's higher than the binding energy of the nucleus it fissions." Ultimately, this is more plausible - but less exciting - than the article's claim that the number of neutrinos is changed as a result of a solar flare.
The best "next experiment" would probably be to set up a continuous, ongoing decay measurement, and compare decay rates to our other neutrino detectors next time there's a supernova. The existing neutrino detectors always pick up a massive burst of activity just before a supernova becomes visible, because the extremely high-energy fusion going on as a star dies pumps out a lot of them. If they don't see the expected changes in decay rates at the same time, whatever process is changing the decay rates is either 1) not neutrino count, or 2) for some reason it's only neutrinos from the sun.