At first, I was overcomplicating it. Then I tried drawing out the pattern, including the shadow effects. That's when it became clear to me.
Arranging the hexagons as they did in a 3D cube pattern has quite an impressive effect. But, the folding of the design, is actually quite basic.
There's really only one technique involved beyond the usual triangle grid.
The last image shows a rudimentary crease pattern. It's simply large hexes all connected by triangles. To fold the design you perform a sort of waterbomb fold on each triangle. That enables you to flatten the hexes. The lines in the triangles that surround the central hexagon illustrate how to fold them. You squeeze together the outer edges of the triangles in mountain folds collapsing along those inner vertexes as valleys.
If you lay out your hexes in a diamond shape, you get the 3D square pattern that's shown in the design.
The first image shows the front back lit. Second is the rear. Third is front unlit.