The article claims that, in the fossil record, there appears to be a period of about 62 million years to major changes in life on Earth, and that this time period matches the period of one cycle in the sun's orbit around the Milky Way.
To go further, we need to understand how the sun orbits the Milky Way. The Milky Way is mostly flat, shaped like a Frisbee or a dinner plate, though it does bulge a little in the middle. We call this Frisbee part of the Milky Way the disk. The sun lies about two thirds of the way out from the center of the Milky Way, and for the most part traces a circle about the middle.
However, the sun also moves up and down, like a bobber in water. This is because the disk of the Milky Way isn't empty, but consists of stars and gas, each of which has its own gravity that pulls and tugs on the sun and our solar system. The amount of time it takes the sun to bob up and down once is about 62 million years, about the same length of time as the changes in life on Earth.
Now, when the sun moves up out of the disk (or "north"), it is moving in the same direction that our whole galaxy is moving. When the sun moves back down (or "south"), it is moving in the opposite direction. The scientists in this study claim that, when the sun is moving north, it moves out of the Milky Way's magnetic field, which helps to shield cosmic rays coming from other galaxies. If this is true, then the increased cosmic rays could spur changes in life on Earth.
Is this true? Maybe, but I remain skeptical. We know very little about the shape of the magnetic field of the Milky Way, and how far it extends into space. We know even less about how much cosmic radiation this magnetic field shields us from. And we don't know exactly how long it takes the sun to bob up and down once, or how much this bobbing might vary from one bob to the next.
But this is how science is done. The new study proposes an explanation for something we observe. Now it needs to make a prediction about something we don't yet know the answer to. If, in the future, that prediction holds up, then the hypothesis that the sun's bobbing motion affects life on Earth is made a little stronger. If that prediction fails, then the hypothesis is less likely to be true.
Once in the past, a scientist named Milutin Milankovich proposed that the Earth's ice ages were due to natural cycles in the exact shape of Earth's orbit. He predicted that, as the ice ages were more precisely dated, these cycles should match ice ages exactly. The idea was radical and not accepted by a lot of scientists until it was found that the timing of the ice ages follows these cycles with very high precision. Now, Milankovich's theory is well accepted by many climate scientists and planetary scientists.
So, it may end up that many extinctions and changes of life on Earth are due to this bobbing motion. Similar hypotheses have been made before, but these were not able to make testable predictions, a vital part of any scientific theory. We shall see!