How does this circuit work with a transitor?
This is really two separate questions:


  Why doesn't the LED burn as the resistor is placed after the LED?


The order of the LED and resistor doesn't matter. They can be in either order. Please see the question Roger Rowland linked to in his comment for more info.


  Where does the base of the transistor get current?


It gets a tiny amount through the finger being held on the green and yellow wires. It will be a high resistance connection, so there will be only a tiny amount of current. The function of the transistor is to allow a much larger current to flow into the collector. Both of these currents then flow out of the emitter, through the LED. If the current gain of the transistor is high enough it will be able to light up the LED.