To get a significant inclination (ie away from the low inclinations of the inner planets) and a drop in the perihelion (ie highly oval rather than the round orbits of the inner planets) you need a close pass of a large body. As the little think comes close to the big thing it is thrown randomly, away from the nicely round orbits of most things in the early solar system. So if you find a body with those two features, you know it passed close to something big somewhere our near the highest part of its orbit. That is the evidence of a large object out beyond pluto.
The clustering of the arguments of perihelia is less straight forward (you can see it in Figure 4 of the linked pre-print PDF in another comment) and my limited understanding is that they are shepherded into that configuration over time by the large body. That is more clear when you see the results of the numerical simulations I suppose.
The paper authors are actually arguing for a Planet X which has had periodic encounters much weaker than that at a distance of a few hundred au.
It's orbital eccentricity is ~0.94!
The amount of sky we have to look for Planet IX is enormous (and given how faint it is likely to be), and the evidence so consists of about ten of these TNOs with these orbits (not many, but it's very hard to make these orbits). But there are potentially hundreds of bodies with orbits like these waiting to be found by a the soon to be constructed LSST, which is an 8 meter telescope that will survey its entire sky every 3 days. With many more of these orbits to predict from, we may really narrow down the search space to look for it, and ultimately test its existence.
[1] Predicted by Konstantin Batygin and Mike Brown, the latter of which played a big part in helping to disqualify Pluto as a planet by discovering Eris. I know from a lunch-talk that he greatly relishs the thought of both demoting the former Planet XI and predicting the new one.