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.

There are spots where a vertex would have two hexagons at it rather than three.

For instance, look at the border between the upper left and lower left islands. In this picture, those two islands are only touching along a third of the border that they share on the sphere. If you look at the leftmost point of where they are touching in this picture, that's a vertex that would have only two hexagons.

There are 6 points on the map where pairs of adjacent “hexagons” are folded in at the corner so that they border each-other along two sides. If you like you can think of these as pairs of pentagons rather than hexagons.