You just pry the rings off with a small screw driver. This model has a large one at one end and a small one at the other. Once inside, you need to cut the old wire off of the switch contact and the bulb socket.
This will make more sense as you see for your self the old wire and what you must do to install new wire in the fixture. Yes, it can be scary the first time you try this. Just remember, somebody, just like you, put this together back in the old days. Pictured below is the new wire soldered to the appropriate pieces.
Installing this back into the original fixture can be time consuming. You have to get the wire path just right so the two halves fit back together snug. It may take several tries. I made up several new 4-letter words while attempting this. I will not share my words with anyone. You must make up your own. I did run into one other problem that was not a normal "re-wire" issue. The center bulb contact in the socket had broken off. I had to make a new one. I folded some copper foil I had and slid it down into the slot and then soldered it in. You should not have to do this but this really shows the "do what you gotta do" attitude you must have in dealing with some of these old sewing machines.
Now for the splicing of the power wire, the foot controller and the light in the motor mount.
All the new wire is in and just needs to be cleaned up, insulated and tucked in. I used 2-wire nuts and a shrink tube case to protect the splices. Shrink tubing can be wonderful by itself in protecting old wire.I used wire ties to keep the wires from pulling out of the case.
The foot controller is the easiest part to re-wire. Most are easy to open up and replacing the old wire is straight forward. I use wire ties here also to lock the wire inside the controller.
If you made it through this much of this post, you may wonder where I get the new wire for the project. I found that buying cheap extension cords at the thrift store is the least expensive. Try to find some that are flexible and not so stiff. Cut the length that you need and leave the plug on for the power end. There should be enough information here for most people to rewire most of the old machines. This Model 66 was different than most since it spliced in the motor mount instead of the standard, Singer, 3-prong jack/plug. The prongs are numbered if you look close inside and are even color coded. Please be careful. I have blown the circuit breaker on more than one occassion. Click on the diagram below for a larger image. The area inside the shaded "Standard Singer Jack" is exactly the same wiring as the above model 66 motor mount splice. There are some other configurations that include a power recepticle where you plug the light into 1 socket, and the motor into the other. They still wire the same.