Notice how the lines are stretched, Not every hexagon in the tiling represents the same area if mapped back to a sphere. The non-polar edges of the tiles pictures are larger. If you actually printed it out, and tried to wrap it around a globe, the middles would have to expand or the edges would have to shrink for the tiles to join.

Edit: Basically he hasn't actually tiled a sphere with regular hexagons (which is what the proof said was impossible. He has titled a flat projection of a sphere with regular hexagons, which would have to be morphed to irregular hexagons if tiling a sphere when the projection was reversed.

Not taking away from the tiling, which is quite interesting in itself.