what is configuration space
It means that those things aren't really fundamental. What's fundamental is some other set of things that aren't too difficult to work with mathematically, but which don't correspond exactly to anything you're familiar with. The best you can do is to add in "probability functions", such that "state of the world X" means that you'd have some chance of detecting a particle here, and some chance of detecting a particle there, and so on.
It's that jump from one to the other that's awkward, and to be honest not 100% understood. But it's actually better understood than a lot of the woo-woo descriptions of quantum mechanics make it out to be.
Now, when I said that the math isn't too difficult to work with, that's a bit of a lie. It's all about Simple Harmonic Oscillation, except using complex numbers. That's really not too bad, but it's definitely new.
And both of those are among the details which can be safely ignored. The only math you really need in order to understand QM conceptually is linear algebra. (And actually, even the harmonic oscillators are not that difficult to understand. If you understand Euler's identity, you're 90% of the way there.)
How is that different from hidden variables?
One way to look at it is that every one of those pieces of configuration space apply to all of space and all of time. That's not forbidden. But it also means that these values aren't directly accessible. They can only be partially inferred.
I'd like to know more about this since I have never heard of this concept before and I feel like it has far-reaching implications
The unobservable parts are not really any different from indirectly inferred concepts like potential energy. It's just that even seemingly concrete notions like location turn out to be macro scale approximations of this phase space. They're not accessible only because you are a macro scale object. You can very much demonstrate that it's real by simple experiments. You just need to be able to take them seriously, which is unexpectedly difficult because classical physics seems so intuitively obvious.
gp's premise about learning with the multi-particle landscape first sort of blew my mind.
Right.
> Is this why string theory requires higher dimensions?
No. The higher dimensions of string theory are actual physical dimensions. Configuration space dimensions are pure mathematical abstractions, and there are potentially an infinite number of them, one for every degree of freedom in the system. Look up "Hilbert space" if you want the gory details.