I think they are just taking multiple images while adjusting focus for every "image" that you take. Tapping the frame just causes a standard contrast-based focus search through the stack. Nice gimmick.
Not quite, it's a plenoptic microlens array camera, similar to a shack-hartman wavefront sensor. The best analogy is it breaks the normal FOV into an array of smaller groups, each of which has its own microlens. What this allows, through processing, is the determination of the wavefront direction, rather than just the intensity. Since there will be multiple wavefronts, deconvolution processing then allows a user to choose the one that best represents the focus that is desired (at any given fixed focus position, different distances of object will result in different wavefront curvatures).
