Understanding Ohm and Kirchhoff
your first question:


  ...
  Therefore, voltage drop is 17.55V.
  
  But what does this mean?
  
  Using the same calculations, voltage drop for the different pieces of the circuit are:
  
  Resistor 1: 0.0045 * 5600 = 25.20V
  Resistor 2: 0.0045 * 3900 = 17.55V
  Resistor 3: 0.0045 * 1200 = 5.40V
  
  And those total to 48.15V, so our math is close but has minor rounding issues.
  
  But, again, what does this mean?


This referes to the Kirchhoff voltages's law. The sum of all voltage drops in series circuit must be equal to the voltage applied (this is not entirely correct, but serves the proposed circuit and avoid complications).


  Does it mean that tapping into the circuit between resistors 1 and 2 would give me 22.8V (48V - 25.2V)?


I fail to understand this question. The voltage drop obtained for each resistor is the voltage that you can measure with a voltmeter conected to the resistor pin's.


  If I have a sensitive electronic component that accepts 3V-6V, would it be safe to draw this power from the circuit after resistor 3?


Definitely not. Any circuit connected to the shunt, completely modifies the behavior of the original circuit, and hence the original voltage drop.


  ...
  It says it can accept 6-20V (but USB is 5V!), and somehow it deals with all of that excess voltage to provide a steady, regulated 3.3V. What dark magic is taking place here?
  ...


The voltage regulator is a retrofitted system, i.e. when a circuit is connected to the regulator's output, an internal circuit operates to maintain a stable output voltage.
The difference between the input voltage and the output is absorbed by the regulator, as having a "variable resistor". According to the connected circuit, the internal regulator circuit "set" the value of this "variable resistor" (actually it is a transistor, for a linear regulator) so that the voltage is maintained within a specified range. 

Voltage regulators are a whole specific field of application in electronics. There are many types. Particularly, the LM317 regulator is a linear regulator. Implements a transistor as a regulating element between the input voltage and the output voltage. The control of this transistor lets adjust its voltage drop, according to the circuit you want to feed.