Why does this not short circuit?
I REALLY, REALLY hate that "electricity follows the path of least resistance" expression, since so many beginners seem to see an "only" in there somewhere.  In fact, electricity follows all possible paths, with the current in each path determined by that path's resistance.  The highest current will flow in the path having least resistance, but current will flow in any other available path as well.

Current follows both paths you show.

R1 and R2 form a voltage divider that determines the output voltage of the LM317.  The LM317 attempts to maintain about 1.25 volts between its output and adjust terminals (pin 2 and pin 1).