Looking at the image the power for the LEDs is coming straight from the DC jack, to the resistors, then though the LEDs. The ground (cathode) comes off each LED then goes to the switch which then completes the circuit via the ground (depending of which poles are making contact) on the DC jack and therefore allowing each LED to work.
To use the three-pin LED you'll need to reverse the wiring. The common cathode pin will need to be wired directly to the ground of the DC jack, the power coming off the DC jack will then need to go to the middle pole of the switch (the blue wire in the photo) and then the pink and green wires going to the each resistor (4.7k is a good value to use if you using a 9 volt power supply) on either side of the LED.
In a nutshell:
Blue wire: To the positive on the DC jack.
Green wire: To the one of the resistors on one side the LED
Pink wire: To the resistor on the other side of LED
Where to green and pink used connect to the LED, connect to negative on the DC. This will be only a single wire as each side of the LED share a common ground.
I hope this helps? And isn't too confusion. Let us know how you get on.