(I am not a particle physicist or a mathematician of the right sort, but I can kind of follow this sort of thing)

Okay, the context is that you've got particles, and they're fundamentally all the same, but they're "turned" in different ways. Think of a ball with 3-color LEDs inside: you can rotate it around three axes, and move it in three directions, and you can also cycle its color and change its blinking pattern. Particles are like that, except that the topology is weird: it's not back to the same orientation until you turn it around 720 degrees, instead of 360 like normal objects. The "gauge group" is the rules for how you can change things. For example, the total color of the universe is white: if you turn something from red to blue, you have to turn something else from blue to red; but you can also create a pair of a green and a purple (anti-green). They write all these rules up in math, and it's tricky because a lot of the features vary continuously (that is, you can rotate something an arbitrarily small amount). And due to the interaction of the rules for one property with the rules for other properties, there are only certain combinations of properties that you can get. They work out all the combinations that you can have and those are what you see as "different" particles that your experiments show. Of course, we don't know what the rules are, and we're trying to figure that out from what combinations of properties we've seen and which ones we're speculating are impossible. And it's hard and takes a lot of calculation to figure out what a candidate set of rules would even mean as far as results. And people are looking at known results and trying to describe them better than "we've done a billion things, and a billion things happened".

Now, the math of rules for how things can interact turns out to be sort of limited; there are basically 4 normal cases, which are boring, and then there are a few exceptional cases, which are interesting. Of these, the hardest to prove stuff about is E8, and it's just now becoming clear what combinations it allows. It's like one of those puzzles where you press a corner and lights change, and you have to turn off all the lights, but it's got dozens of corners and dozens of lights and every time you press a corner a bunch of things change at once, and there are different kinds of corners and it also matters exactly what angle you're holding it at, so there are hundreds of things you can say about each move.

And the mathematicians working on E8 recently said, "well, you can get positions like this and not like that", where "this" and "that" are big complicated lists. And this physicist read that paper and said, "hey, those lists are familiar; I made similar lists of particle interactions". So the proposal is that particles work like E8 in what kind of rules they follow. And it's a really nice theory, because E8 is essentially the most flexible set of rules you can have without it falling apart into just anything being possible (and some rules or properties just not mattering).