The resistor is needed to limit the current drawn from the Arduino output pin to below is absolute maximum rating of 40mA. A normal bipolar transistor presents a low resistance between base and emitter once the voltage exceeds about 0.65V. For a darlington like the TIP121 it is more like 1.3V. The Ardiono output pin provides up to 5V. Without the resistor, the current will be limited only by the internal resistance of the Arduino output (which is a few tens of ohms), and the current will be much greater than 40mA.