The brittle wire has likely been 'cooked' by the can light. The heat drives off the volatiles that keep the insulation flexible. This is especially common with older wire. It will often be only a few inches of wire near the lamp.
First: Here in Canada there is a series of books, "{province name} electrical code simplified" This gives you the relevant chunks of the code with reasonable explanations both for why and how. Find the equivalent book for your jurisdiction. I recommend this class of book to anyone who wants to be their own electrician.
Second: Turn off the breaker that affects the lights, and rummage around. You should find a junction box beside, or above the can. It may help if you look at some cans at a Big Box store so you get an idea of what you are looking for. With most cans there is a cover ring that spring clips to the can itself. These vary in how they attach. Play and wiggle. Get the model number (should be on a sticker inside the can) and see if there is a web page about it.
Often cans are attached to some wooden part of the ceiling. (There are retrofit ones that attach to ceiling drywall.) Look for nails or screws inside the can. In some cases the junction box is built into the can.
Sometimes the attachment cannot be accessed without taking down part of the drywall around it. This is when I would consider the professionals.
It might be possible to 'reinsulate' the brittle ends of the wire. No idea if this is code or even possible to do.
Third: Now you have to decide to do one of several things:
- Replace the can. If you can get the cover ring off you may be able to see if removal is possible. You can remove less than cover ring width of drywall and still have the cover ring cover the gap.
- Replace the wires between the junction box and the lamp holder in the can. This may not be possible if the wires are molded into the lamp holder.
- Replace the wires and the lamp holder in the can. This will often allow you to replace the high temperature incandescent fixture with a much cooler and more efficient LED fixture.
Myself, at this point I would look at replacing the whole can with a smaller LED set up. This may allow insulation where before there was space, which could be a significant change in a exterior cathedral ceiling in a cold climate. (Most cans have to be in a box so that the heat from the lamp can dissipate. Don't know if the code as caught up to 3-5 watt lights yet)
You also have a meta decision: Do you want to tackle this yourself, or hire a pro to do it.