I finally found the answer at
this link and you can also buy them on that website. They're called plastic edge protectors, and they're for covering the sharp edges of welded wire to prevent injury. This makes sense - apparently rabbit cages are usually made by cutting pieces out of bigger sheets of wire and then fastening them together somehow. This is similar to the way I built my aviary, and I can tell you from experience that when you trim a piece of wire it leaves a little nub behind that is sharp. I designed the aviary so the sharp little nubs are on the outside where the birds can't reach them, but they can still be a hazard to me sometimes lol. It's possible to file the nubs down, but this is time-consuming and there are way too many of them on an aviary to make it worthwhile.
Apparently it's fairly common to use these edge protectors with rabbit cages. Here's a website with a bunch of different cages and they all have this edging around the door:
http://www.martinscages.com/products/cages/rabbit/Here's a picture from a different website that shows what it looks like with the door open. You can see that there's nothing on the side that has the hinge, but the other three sides have the edging. The actual website is
here:

There are several companies selling parts for rabbit cages online, and several sites with how-to information. This might be a good source of ideas for cage-building, and the parts you need for the cage.