Different ways of pin access
Why is direct access to pins by the PRU, so much faster than
  memory-mapped IO way?


The Beagle-thing has a regular processor and two fast CPUs running at 200 MHz. Theynare called PRUs and the document you linked says this about the PRU (quite close to the top): -


  It is an independent CPU with its own memory and instruction set. It
  can run its own program, completely independent of the Linux kernel on
  the main CPU. It's fast (200MHz clock), all the instructions take
  known constant times and you have it all to yourself, so you can use
  it for things that require a hard realtime response. The 'bone has two
  PRUs.


So you can use fast IO from the PRUs or slower IO that is memory mapped to the main processor but the main processor is slower because it's running the linux operating system.

The PRU IO is likely faster because hardware wise the pins connect directly to the CPU chip whereas memory-mapped IO (from a labouring CPU running linux) might only talk to its IO chips every few milliseconds.