Is it possible to explain what’s being claimed here in terms even more basic than those used in the article itself?
They don’t say that higher-than-3D materials can be made. They say that, judging by these results, it might be possible to make systems that behave in physics experiments as if they have topology that is higher than 3D.
Well that clears it right up.
If you perform a Fourier Transform on that post, it becomes perfectly clear. You’re reading it in position space instead of momentum space.
I’m far from an expert on this kind of thing, but maybe I can get some of the flavour across. So, let’s first look at the concepts used. A topological insulator is a particular material which is an insulator in its interior, but has conducting states on its surface. The word ‘topological’ essentially comes from the fact that the effective theory describing this material is a so-called topological quantum field theory, but that doesn’t really matter. What matters more is that such a material can’t be described within the traditional ordering approach to matter in different phases: a liquid is disordered, while in the transition to a solid, the ordering of a crystal lattice emerges. These two states of one and the same material differ only in its ordering, which then must be the cause of the differing physical properties.
But this description is insufficient for a topological insulator – it possesses a new kind of order, called sensibly ‘topological order’. This is what makes this surface-conductor bulk-insulator behaviour possible.
Now, what they’ve found is essentially that a certain 1D quasicrystal (which material is again characterized by a particular order that is non-periodic, i.e. does not repeat itself, like all regular crystals do) can exhibit properties very much like a 2D topological insulator. What that means is (I’m guessing somewhat here) that the mathematics used to describe electrons in a 2D topological insulator applies to the properties of the photons in their quasicrystal.
This is not actually that unusual: everything under the umbrella of ‘holographic principle’ is basically similar. The most well-studied example is what’s known as the AdS/CFT duality, in which a certain quantum field theory (a conformal field theory or CFT) is mathematically equivalent to a gravitational theory (like a string theory) in a higher dimensional (AdS) space. Roughly, a bound state of the quantum theory of a certain size r is dual to a localized particle in the AdS-theory at the point z = r, where z is the ‘extra’ dimension. Thus, the scale parameter of the quantum theory gives rise to the additional dimension – smaller things are farther away, nothing new about that. 
TL;DR version: think of it like a hologram. You can encode 3D information on a 2D surface; they’ve basically found information describing a 2D topological insulator in a 1D quasicrystal. The same thing should in principle be possible in higher dimensions.