I can summarise the basic gist of the paper in relatively non-technical language (other people, please comment if you disagree with what I'm saying here):

Einstein's equation says that the curvature (read geometry) of spacetime is equal to the stress-energy tensor, which basically measures how much mass/energy/momentum there is in a place at a time. However, in quantum mechanics, the universe is in a superposition of states with different distributions of mass. A theory of quantum gravity would therefore say that the universe must therefore be in a superposition of states with different geometries. The alternative is to have semi-classical gravity, where there is only one geometry of spacetime.

The most obvious way to construct a theory of semi-classical gravity is to say that the geometry of spacetime is actually related to the average distribution of mass of all the Everett branches (if you're an Everettian). To test this, you can take a large mass, and put the universe into a superposition of two states: one where you move the large mass to the left, and one where you move the large mass to the right. You then see if your mass is attracted to the mass in the other Everett branch. If semi-classical gravity and the Everett interpretation are right, then both masses should curve spacetime, and that curvature of spacetime should be felt by both of them, so each mass should be attracted to the one in the other Everett branch. If semi-classical gravity or the Everett interpretation are wrong, then the masses shouldn't feel attracted to the ones in the other Everett branch.

The people who wrote this paper did something that was essentially equivalent to the experiment described above, and discovered that the mass was not attracted to the one in the other Everett branch, meaning that semi-classical gravity and the Evererett interpretation can't both be true. They also argue that semi-classical gravity implies the Everett interpretation, and that therefore semi-classical gravity can't be true, although I am suspicious of this argument.

Note that it is possible to do tests for semi-classical gravity that don't rely on the Everett interpretation, although these don't seem to have been done yet (see http://physics.anu.edu.au/projects/project.php?ProjectID=31).