Try this: when we look at a galaxy, we can see how fast it's rotating. We can do this using spectrometers that can see where light from stars is red-shifted (which indicates those stars are away from us) or blue-shifted (moving toward us).
The faster a galaxy is rotating, the more massive it must be, because without that gravity the stars would just be flung out into space like toddlers on a malfunctioning carousel.
We can also make a pretty good guess at the amount of "normal" matter in a galaxy (i.e., not dark matter) by measuring how bright a galaxy is, since nearly all light comes from stars, which are, as far as we know, composed entirely of normal matter.
So, putting those two things together, we encounter a mysterious thing: having observed thousands of galaxies, nearly all are rotating much faster than is expected given their estimated mass. One possible explanation for this is that there's another kind of matter in there that doesn't emit any light, which we call dark matter. To be more precise, not only can this dark matter not emit radiation, it cannot interact with radiation at all -- it can't absorb or reflect it. Otherwise we'd see it, just as we see clouds of dust in galaxies (which is accounted for in our mass estimates).
Other proposed explanations involve changing the way gravity works. These are very interesting and compelling ideas, however this newly discovered galaxy would appear to weaken the case for modified gravity theories, because under those theories we would expect this galaxy to be rotating as fast as all other galaxies, since gravity is a function of mass. But that's not what we see; this new galaxy is rotating much slower. It is, in fact, rotating at the rate we would predict if it didn't contain any mysterious invisible "dark" matter. Which is why we think it doesn't!