The Arduino does not get 5V power from the TIP-120 transistor.
There are four ways to provide power to the Arduino:
USB cable provides 5V 500mA.
Built-in regulator can accept 7V to 12V and provide 5V 1A.
There are two ways to feed the built-in regulator:
The power jack (center + and sleeve -)
The Vin pin (+) and Ground (-).
You can use an external regulator and provide regulated 5V to the 5V pin (+) and Ground (-).
The transistor acts as a switch: http://www.electronics-tutorials.ws/transistor/tran_4.html
The Arduino can use a weak (40 mA max) 5V output connected to the Base of the transistor to switch a higher current and higher voltage.