I'm not sure where they are going to but the connectors I see on the led are either the screw in type like a bulb or a connection with a pin that's different from the pin connection I currently have with the halogen bulb.
CV is right. You'll have to do 2 things here
1) Confirm that the wires in your picture carry 110/120V [It's not likely that those are low voltage but I'd use a meter or a Klein tester to be 100% sure]
2) Splice in the LED Retrofit
Let me be clear that I am not an electrician, so you want to consult someone about this suggestion to make sure this it up to code.
The suggestion would be to cut off/pull the wires out of the circle plate, as that plate is part of the halogen fixture. Then you'd splice in a bulb socket (buy in HD) and attach the retrofit. Retrofits come with 2 connectors (I've bought/installed many dozens)- a bulb socket connector and a quick release connector (so that you can screw the socket connector into place while the LED is disconnected). If you want to skip buying a bulb socket you could (in theory) cut the socket connector off of the retrofit and splice to that wire instead (while still preserving the quick connect release), but that's a modification to the fixture so you have to make sure it's up to code to do so (would be cleaner and simpler but safety code is a higher priority).