I will give it a try.
First you don't need all three colors of the wires going to the switch.
Take the black wire( we will call it common) it will run from your power supply to your turn out motor. (Black wire) Now take the other power supply wire and take it to one of the center terminals of your switch.( NOTE you have a double pole switch so you will only use one side for the turnout control, the other half of the switch could be used for signal controls)
Now on the same side as you hooked the power from your power supply( the center terminal) you hook up the green on one side and the red on the other. When you move the switch lever from one side to the other it will power up turnout from one or the other, and it should move your turnout. If the lever on the switch does not match the position of your turn out just swap ther red and green wires.
Hope this helps