The resistors between the Arduino and the 9 LED anodes will set the current for each LED, protecting both the LED and the Arduino pin.
The Base-Emitter junction of a transistor acts like a diode with Base the anode and Emitter the cathode. Without a resistor that 'diode' would short the Arduino pin to Ground and draw too much current. The Base-Emitter current is multiplied by the transistor so you don't need much. The "beta" of the transistor (DC Current Gain) is how much the current gets multiplied. You divide the Collector-Emitter Current (total through 9 LEDS) by the beta to determine the minimum current you need going into the Base from the Arduino pin. Any current above that and well below 40 mA (the pin maximum) will do. Typically a 1K resistor (about 5 mA) will work for all practical purposes.