A galaxy (elliptic, or spiral) is made out of billions of stars, like our sun.
These stars rotate around the center of the galaxy (very slowly, think millions of years for 1 rotation).
A rotation curve measures the velocity of stars as a function of distance from the center of the galaxy.
Newtonian physics (or Einstein's GR) says that the rotation curve should decay with distance, ie. with greater distance the stars' velocity should go down --- assuming the matter in the galaxy is the visible matter that we see, ie. the stars (which shine light).
The problem is, there is a rich set of observational data, from many different experiments, telescopes, and methodologies that show that the rotational curve is in fact flat, it does not decay.
There are 2 big competing theories to explain this discrepancy:
1. Assume that there is a lot of unseen, non-shining, ie. Dark Matter (DM) in the galaxies (also ours). If you put the appropriate amount of dark matter in there, with the right distribution, you can reproduce the observed rotational curve. There are also other places is astrophysics/cosmology where having dark matter (specifically Cold Dark Matter, CDM, where cold just means "slowly moving") is useful. The biggest example is to explain the history of the Universe and the observed Hubble-constant. In fact the standard model of cosmology is called λCDM, CDM for Cold Dark Matter (λ for the cosmological constant, currently modeled as Dark Energy, not relevant for this discussion).
2. Assume that Newton was wrong and gravity is not exactly 1/r^2 --- this is called MOND, Modified Newtonian Dynamics. This way you can also reproduce the observed rotation curves. This is much less popular, because: (i) physicsts don't want to give up the beautiful and geometric simplicity of 1/r^2 (ii) Dark Matter is also useful for solving other discrepancies in astrophysics/cosmology.
What this article is saying is that, even in the first Dark Matter model, per the model DM distributions inside galaxies that also work with all the other places where DM is used to explain something (eg. in cosmology), at some distance from the center, the dark matter bubble has an edge and stops --- and then the velocities should finally break down. However, these latest observations are showing that the velocities remain constant even beyond the modeled/assumed DM bubbles. This is an additional ε argument in favor of MOND, and science proceeds.