Looks to me like you will have to loosen the adjusters off and see if that gives you enough slack to rotate the cable around to the slot to slip them out, if you don't have enough slack you will have to take the handle bar throttle housing apart and release the cables from the from the throttle hand grip, that will for sure give you enough slack.
Be careful when you take the throttle hand grip apart, there are small pieces in there that like to jump out and roll into the farthest reaches of your garage.

 

As HOTLAP said you need to create as much slack at the grip adjusters as you can ... you should then be able to remove the cables from the carb ...

 

Yes, I found my little adjuster shoe months later. My little star wheel broke off so it was useless anyway.

John

 

 

Have a look on you tube, Moonfleet41 has good Sportster service vids.


Dave 04 XL1200C

 

Looking at the first picture once again does that bracket that holds the two cable sleves happen to have nuts or bolts holding it on to the carb that are out of view in the picture? if so removing them would allow you to move the bracket over and down creating lots of slack.

 

Don't forget, you can manually open the carb and get more slack on the throttle cable, then remove the idle cable.

John