The major problem with understanding articles like this is that while it typically doesn't involve quantum entanglement, it's close enough to quantum that it makes the science writers get all giddy about the words they are throwing around and they do their usual "why inform the reader about what is going on when we can just make them go Gee Whiz" schtick.
The key word is "quasi-particle" which is somewhat less exotic than it sounds. It is a combination of what you might call real or normal particles that produces some sort of pattern in it that itself acts like a particle of some sort. The resulting "quasi-particle" can have all kinds of interesting properties that normal particles can't have on their own, but what makes them "quasi" is that they can't exist on their own. They're intrinsically on top of some substrate of normal particles.
One of the simplest quasiparticle is the "electron hole". Take a lattice of some electrically neutral substance. Remove one electron from it. There is now an "electron hole" in it. You can treat that hole like a particle now. It can "move" to another location by having the real electrons change places. It can "flow" through a series of such events. You can model a lot of things with "electron holes" that act in very particle-like ways. But they don't exist on their own. This one is simple because you don't even need quantum mechanics to get a hold of it in your head.
Many more complicated scenarios are possible. Many interesting things can happen with them. Most, if not all, news articles about "new phases of matter", which science writers love to write about only slightly less than making "woo woo" motions with their fingers while talking about quantum entanglement, are new quasiparticles of some sort. This is somewhat less interesting than they think because if you include quasiparticles as "phases of matter" then there are already hundreds or thousands, but the science writer wants to write an article about every single one of them as if the list is now "solid, liquid, gas, Weyl semimetals" and then write the next article as if the list is now "solid, liquid, gas, ELECTRON HOLE" and so on and so on for each new quasiparticle.
But from this perspective, the list hasn't been so short as "solid, liquid, gas" for well over a hundred years now, and while adding a new one is often good science, it has also been "just" another one of thousands for a while now.
This post is not an explanation of "spin ice", "Wely fermions", or anything else; what this is is the "secret decoder ring" to remove the wiggly fingers and the "woo woo" noises the science writers add to this topic every time they write about it and to give you the terms you can Google and start reading up on what is one of the most interesting and productive fields in the hard sciences right now. Everyone loves to talk about how stuck particle physics is, but physics is making a lot of interesting findings in the field of making the particles we know about sing and dance in all sorts of new and interesting ways.