How to wire external power supply on stepper motor
All you have to do is supply 12V to pin 10 of the ULN2803 and the common wire of the stepper motor (the 5, red lead here). That is the reason the ULN2803 is used. It is simply an array of 8 Darlington pairs of BJTs with open collector output. The schematic of one pair is on the datasheet, page 2.

The Darlington pairs are used instead to one BJT to amplify more current. The amplified current from one BJT is fed into the base of the next to amplify even more. This configuration allows much current to be sunk from a small input source.

The open collector outputs allow the smaller logic input to be interfaced with larger supply voltages. Each pair of transistors will simply sink the load to ground, effectively turning it on. The downside is you cannot push a logic high to the load, but that isn't a concern for this situation since there is a common supply and each coil in the motor is energized by being sunk to ground.

Pin 10 of the ULN2803 is connected to the cathode of 8 diodes whose anodes go to each of the 8 outputs. These diodes are referred to as clamping diodes. When your supply is connected to this pin, it will prevent any of the output pins from rising above the supply voltage. This a useful feature for switching, inductive loads such as stepper motors that produce back-emf.