Why add a diode to a transistor controlled DC motor circuit?
See The flyback diode and its applications


  Is there something about the rotating nature of the DC motor that causes reverse current or something, and the diode is there to protect the arduino?


Essentially yes, but due to the coil nature of the motor. The same property applies to solenoids such as relay coils. Because it's a coil, it has a high inductance. Inductance limits the rate of change of current flow. The diode allows the current to flow in a loop through the diode rather than trying to force it through a nonconducting transistor.