How do Hoerner Wingtips work?
How exactly does acceleration of air occur just by having a convex
  underside


Just like the curved forward surface of a wing does it. Imagine for a moment that the air flows along a straight path: Now it would "see" that the surface below it curves away from its path of travel, and if that path would remain unchanged, a vacuum between the wing and the air would form. Reluctantly (because it has mass and, therefore, inertia), the air will change course and follow the wing's contour. This requires lower pressure, to make the molecules overcome their inertia and change direction.

Near the tip the flow has a sideways component, flowing away from the center on the lower side. This sideways flow, coupled with the sideways curvature on the lower wingtip, creates suction. This area of lowered pressure will suck in more air from inside. In essence, the curved lower side of the Hoerner wingtip extends the suction which is found on the upper surface of the wing to the side of the wingtip.


  As we can see, the effective wingspan using Hoerner Wingtips can be
  increased without increasing geometric wingspan.


Well, that is at least what Hoerner claims. Maybe it does move the peak suction of the wingtip vortex a little outside, but this has no effect on the flow field behind the wing and the eventual location of the trailing vortex. Below is a graphical overview of different wingtip shapes and their sideways location of the core of the tip vortex. 

Wing-tip shape and tip-vortex location for a family of wings (picture source)

However, this is not relevant for induced drag. In the end, the strength and width of the rolled-up vortex sheet behind the wing indicates the amount of induced drag. Induced drag itself is the horizontal component of the resulting pressure forces on the wing (see here or here for more on that). The tip vortex is sucked into that vortex system regardless of its initial location. See below for an illustration:


B-747 with contrails (picture source)

You can see that the outer contrails of this Boeing 747's engines wrap around the contrails of the inner engines. This shows how the air is pushed down in the wake of the wing and that the centers of the vortices are slightly inboard of the contrails of the outer engines.