I found the following in posted material for a course at Stanford. It's a pretty intuitive explanation of how alpha shapes work, and clears up my question, if it is accurate.
As mentioned in Edelsbrunner’s and M ̈ucke’s paper , one can intuitively think of an α-shape as the following. Imagine a huge mass of ice-cream making up the space Rd and containing the points S as “hard” chocolate pieces. Using one of these sphere-formed ice-cream spoons we carve out all parts of the ice-cream block we can reach without bumping into chocolate pieces, thereby even carving out holes in the inside ([Ed. the ice cream spoon is allowed to travel freely through space, even to within areas girded by the points S. However, it is allowed to draw a scoop if and only if the points in S can be excluded. ]).
We will eventually end up with a (not necessarily convex) object bounded by caps, arcs and points. If we now straighten all “round” faces to triangles and line segments, we have an intuitive description of what is called the α-shape of S...And what is α in the game? α is the radius of the carving spoon.