This is graduate level physics and math here. It's really advanced stuff, and I'm not sure there's any sort of ELI5 (or ELI18) material out there that doesn't cut out the real juicy and interesting parts.

You're definitely right, but I'll try anyway, for OP's benefit.

Essentially, any particle accelerator in operation attempts to accelerate particles to higher and higher energies. To do this, they use magnets to manipulate the beams of particles.

While we're talking about particles approaching the speed of light (the LHC, for example, gets proton beams up to something like 99.999999% of the speed of light), I wouldn't get excited about this being anything like a "warp drive" technology.

For one, this is an entirely different means of accelerating the particles; the LHC uses superconducting magnets to accelerate the beams where any sort of "warp drive" would ostensibly manipulate the very fabric of space-time itself. (I should note very carefully that the former is very real and used everyday while the latter falls squarely within the realm of science fiction; I'm simply entertaining the idea that warp drive might be possible and saying how very different such technology would be from that used at a collider like the LHC.)

Secondly, just because we're at 99.999999% of the speed of light at the LHC does not mean tomorrow we'll have some sort of breakthrough and cross over (or even reach) the speed of light. Because of the way relativity works, speed asymptotically approaches the speed of light as energy increases; put another way, you can put in an infinite amount of energy and get arbitrarily close to the speed of light, but you'll never reach it.

If you're looking for something to get exposure to some concepts, especially something in astrophysics/cosmology, I recommend something like The Universe, which used to air on the History Channel. This was what I watched growing up, so I'm sure there are analogous programs available now, but something like this might be a good place to start. Bill Bryson's A Short History of Nearly Everything is also a good book appropriate for a wide introduction to some interesting science, also; it gives an overview of a lot of interesting topics without bothering with the mathematics. I found it a fun read.