How does current flow through this Arduino circuit?
To know how things are connected on the Arduino itself you have to take a look at the schematic. This one is the Arduino Uno's. At the bottom right you can see how the board's I/Os are connected to the microcontroller:  

  

So following the line from pin 13 you go to PB5 of the microcontroller. That means bit 5 of port B. If you set that pin to output and make it high you're supplying 5 V to the LED circuit. The series resistor will limit the current to about 5 mA, which is an OK value for the microcontroller.  

So, since the 5 V comes from the Arduino's I/O pin the 5 V wire on the breadboard isn't needed.