Think of a Transistor as a switch. It has a Base, a Collector and an Emitter. When you supply current to the base, current is allowed to flow between the collector and the emitter. So basically, the transistor is working like a temporary push button switch.
The best part about transistors is it will allow you to switch higher voltages on and off. If you've got a motor that you want to run, the Arduino's 5v output is usually too small. Use the Arduino and a transistor to to switch on a 12v power supply to the motor and you're off!