What is the purpose of the elements of this antenna?
It seems to be a patch antenna array, analogous (among many images you could find googling around) to this one:



Patch antenna arrays are a special kind of microstrip antennas. You can find many details about them in this article.

A relevant excerpt:


  PLANAR MICROSTRIP ARRAYS
  
  Planar microstrip arrays are used to form a pencil beam and array elements can be fed in a variety of ways. In the first example the corporate feed is used to activate every array element separately. An example is shown in Figure10,4 which consists of patch radiators and a network of microstrip feed lines. The patch feeds are inset to achieve a good match to the feed lines. The feed lines are made of similar length so the beam pointing is broadside to the array at all frequencies. The bandwidth is limited by the patches themselves, which is typically a few percent. Feed lines are rather long, adding to loss mechanisms, and spurious radiation is caused at bends and junctions.
  
  


Since you said you are somewhat knowledgeable of antenna theory, you might appreciate how such a planar antenna array may generate a useful radiation pattern, as shown by this image of a simulation:



EDIT (After a more thorough search on the Internet)

You may find the following article very interesting, since it is a concise and clear explanation of the basics on patch antennas:


The Basics of Patch Antennas, Updated; By D. Orban and G.J.K. Moernaut; Orban Microwave Products


Some excerpts:


  Topics include principles of operation, impedance matching, radiation patterns, circular polarization, bandwidth, efficiency, alternative feed types, stacked patches and higher mode behavior.


[...]


  Properties Of A Basic Microstrip Patch
  
  A microstrip or patch antenna is a low-profile antenna that has a
  number of advantages over other antennas:  it is lightweight,
  inexpensive, and electronics like LNA’s and SSPA’s can be integrated
  with these  antennas quite easily. While the antenna can be a 3- D
  structure (wrapped around a cylinder, for  example), it is usually
  flat and that is why patch antennas are sometimes referred to as
  planar antennas.