Please forgive my ignorance, because I still don't get it.

Can you give me an example of how this would work in practice? What two assets would you swap in this manner, for example, and why wouldn't that work with traditional financial instruments?

Well it would work with traditional financial instruments. But what wouldn't work with traditional financial instruments is an ability to integrate with them in a matter of, literally, an hour or two. These systems are open and nobody can be stopped from inter-operating with them in any way their imagination suggests. If you wanted to write a bot that does arbitrage between Yen and USD you would be able to do it, easily actually.

If we are going to talk "decentralized finance" - I agree it is weird and dangerous, but I see it a proof of concept. You can get a collateralized loan in seconds, even if you live in a country where banking is almost unheard of. Yes, there need to be things of value that you own on the blockchain but value tied to physical world is becoming more of a thing. And, again, yes it's dangerous and untested. But these are the hacking and experimentation years. One must be able to crawl before one can run.