It looks cool. ;)
And if it's breaking at the joints, you're nicking it when you strip it.
That's it, right? That it looks cool?
And it wouldn't nick if I could strip it with a regular wire stripper! Is there a trick to stripping this junk (without nicking it) besides not cutting all the way down?
Lol. Sorry, you're cracking me up. I actually have some green-cloth/pvc insulated wire in 20 ga. solid that I got off a fella in Tenn. I haven't had any problem stripping it/nicking it, whatever. Like Sluckey says, use good strippers. I like Klein from Home Depot myself. The wire I have I just use on the heaters (without twisting).
"LOL, Sorry, I'm laughing at you." That's all I'm hearing. Redundant and unnecessarily rude post.
That's it, right? That it looks cool?
Isn't that why you bought it? Lot's of people like the way it looks. And lot's of people have bought the stuff you have only to find that it's hard to work with. Most end up throwing the wire in the trash and getting the good stuff. Hoffman sells the stuff you want. It's cloth over wire, no plastic. You don't even have to strip it. Just cut to length and push the cloth back to expose the bare wire. Never any nicks! Very easy.
Is there a trick to stripping this junk (without nicking it) besides not cutting all the way down?
Using a good quality stripper with sharp blades is the secret to stripping any wire without nicking the conductor. Cheap V-notch type strippers can't strip anything without nicking. These strippers work very well on Teflon and should work on cloth/plastic wire too.
https://www.amazon.com/Klein-Tools-6-Inch-T-Type-Stripper/dp/B06WD5737T/ref=sr_1_21?ie=UTF8&qid=1504266770&sr=8-21&keywords=klein+wire+stripper
Or, much cheaper at Home Depot...
http://www.homedepot.com/p/Klein-Tools-Wire-Stripper-Cutter-11046/100630720
So, if you want to use that wire get some good strippers and make some test strips until you learn how to strip cleanly. If you still can't strip without nicking the wire throw it away and get the good stuff.
This wire actually came as part of my weber kit. Since there were two regular push back coils and a pvc-insulated cloth-covered coil, I was earnestly asking (despite my snark) if there was a practical application for the difference.
About the klein stripper: DUH! I totally spaced on this. I have been keeping mine around mostly for crimping and bolt cutting duties since I switched to the GB GS-394 pictured below.

I love love love this tool which I've been using to build synthesizers and other audio circuits for years.
I was using a scalpel to remove the insulation, as that seemed the fastest cleanest way to me, having totally spaced on the Klein tool that's been sitting on my desk since forever, right in front of this project.
Thanks for the clarity, for this ailing space cadet, Sluckey! Your generosity of time and energy continues to impress me.