There's a lot of confusion about what this means, but to be clear: this has nothing to do with ghostly 3D things floating in a surrounding room.

What it's saying is that the 3D nature of the universe might be only approximate. Let's say you (somehow) come up with a two-dimensional universe and physical laws, in which you can mostly accurately (but not completely) calculate the ongoing evolution of a 3D universe. The "mostly accurately" part translates into a slight blurriness, a fuzziness of the 3D world, but it occurs at such small scales that nobody will notice.

Such models have been created theoretically - not long ago some bright spark concocted a ten-dimensional universe that had relativity and spatial deformations and whatnot, but which was mathematically equivalent to a one-dimensional universe that did not.

This experiment is looking for the blurriness.

Now, the story of how this got started is fascinating. Some other bright spark was investigating entropy (chaos), and in particular was interested in the maximum amount of chaos that could be contained in a three-dimensional volume. In a sense, this is like asking the maximum information density of a volume.

Somewhat bizarrely, the equation for the maximum entropy is proportional to the surface area of the volume. This is really weird, and important. The maximum amount of information you can cram into a space is limited by the space's surface area, not its volume.

The implication of this is that you could characterize the entire state of a 3D volume with a membrane. This has been proposed as one solution to the black hole information paradox - black holes are a place of no return, and so they seem to violate the law that information (like energy) can't be created or destroyed. The solution is this: as particles enter the black hole, you get tiny peturbations (bulges, dimples, ripples) in the black hole's event horizon. The idea is that the entire state of that particle retained in these peturbations as they play across the event horizon. The information isn't lost, it's just encoded in this 2D form.

This leads to the startling idea that the peturbations as they evolve are actually modelling the ongoing state of the interior of the black hole. Modelling.. calculating.. simulating. The peturbations on the event horizon are a 2D calculation of the state of a 3D volume.

This is the holographic theory - what if our entire universe, despite its apparent 3D nature, were in fact equivalent to a 2D simulation.