So, first thing, I'm not a astrophysicist nor even a physicist of any type, so I might be misunderstanding things here, but this is how I interpret all this. Hopefully if I've gotten something significantly wrong someone will correct it.

General Relativity matches observations to a point. The issue is that it stops matching observations once you reach galactic scales. In order to explain why that doesn't work you need to start hand waving, and the start of that is dark matter. MOND was thought up not so much as an alternative to General Relativity but as an alternative to dark matter. It tweaks some of the math used in General Relativity to assume that gravity behaves differently at different levels. Basically once you have a strong enough gravitational field it behaves like the gravity we know, but until you hit that point its effects diminish at a different rate. Doing that explains why galaxies behave like they do. For the bulk of the galaxy gravity is strong enough that it behaves exactly like General Relativity says it should, but out near the edges of the galaxy gravity has grown weak enough that it behaves differently. It's sort of hand wavy and leaves a bit of a bad taste in the mouth since there's no real explanation of why gravity should behave that way. On the other hand it doesn't require some phantom matter that we have no observational data to back up.

Either theory falls far short, and both of them require a lot of fudging around the edges to align with galactic scale observations, although MOND once you get past the arbitrary change to gravity seems to require less hand waving. Importantly for the linked paper it also seems to line up with the proposed theory and predict the kind of void the paper is predicated on which would be a strong point in favor of MOND.

Of key point to the proposed theory, General Relativity predicts that in the first moments after the big bang that the universe was essentially uniform, that everything spread out more or less evenly, and it wasn't until much later when things started to form the likes of planets and stars that we started seeing significant variation in matter distribution of the universe. MOND on the other hand allows for variation in that initial expansion. That's important for the paper because there simply isn't enough time in the General Relativity model to explain a void the size that their theory predicts would be necessary to form. MOND allowing for more variability early on on the other hand does allow enough time that a void of the necessary size could exist.

Basically General Relativity on its own doesn't work for things galaxy size and bigger. MOND on its own doesn't work at galactic cluster levels and above. The theory proposed in the paper could explain the discrepancy we see in the rate of expansion of the universe, but doesn't seem to be possible under General Relativity, but is possible under MOND. Both General Relativity and MOND rely on the presence of things not observed yet in order to match with our observations once you scale things back far enough, and neither on its own can explain why the universe seems to be expanding faster than they predict it should. The paper proposes one theory for that, but it's only possible with MOND.