I dont't get it. In the picture, each tile is composed of 8 identical quadrilaterals. So why don't these quadrilaterals constitute a simpler shape that can tile a wall and never repeat?
Of course individual tiles will repeat, but never in an infinite periodic pattern.
Edit: a novelty of this paper is that their shape is "truly" aperiodic, which means no matter how hard you try, you will end up with aperiodic tiling. Existing one-shape aperiodic tilings had to add constraints on how to put two shapes next to each other to ensure aperiodicity.
What does this mean? How can an individual tile "repeat"?
It's very probable (I did not read the paper, only their website page) that the tile only occupies a finite amount of orientations in the tiling and therefore at least (and probably more if not all) one orientation will also be present an infinite amount of times.
However this does not imply periodicity of the tiling.