What is the use of the transistor in this circuit?
The 7805 regulator is setup as an adjustable regulator as adjusted by the 10K pot. The transistor is connected as an emitter follower to increase the load current available to the LED load. Supposedly the 7805 is not able to supply the load current needed for the LEDs and so the transistor acts as a current booster. 

This circuit has a fundamental problem in that there is no current limiting for the LEDs. Normally this current limiting would be supported with some resistors in series with each LED string. 

There is also the additional problem that each of the three LED strings will have a different total LED forward voltage drop. The string with the lowest voltage drop will try to take most of the current from the transistor. This can lead to brightness variation between the LEDs. Once again this problem is solved by having resistors in series with each LED string to balance the current in each string.

BTW this is typical of crap circuits you find published on the web.