I'm not looking for the generic "which way information destroys wave coherence" or "information travels back in time" answers. I'm not saying those concepts might not be in a very broad sense true, but i feel like they're far from complete. What are some of the current theories that try to explain what happens at a deeper level?
I think this a good read: http://jamesowenweatherall.com/SCPPRG/EllermanDavid2012Man_QuantumEraser2.pdf
There is a fallacy when we usually think of the quantum eraser, especially the delayed choice quantum eraser: we know the wavefunction doesn't collapse until it's measured, but for some reason we think there should be which-path information before our particle hits the detectors.
We forget that before the 'particle' is detected, it isn't really a particle yet (just an abstract quantum object we describe with a wavefunction), so it should pass through the double slit without explicitly passing through any one path.
After all, that's what we expect from the double slit experiment, right? If the wavefunction hasn't collapsed yet, the 'particle' displays an interference pattern because its wave-like nature allows it to diffract from both slits at once.
So by making the choice of adding detectors later, we're not actually retro-imposing which-path information. Regardless of whether we erase that choice or not, we never actually change where the 'particle' passed through before it was detected. We only change how the wavefunction evolves from that point onward, which is pretty much what you expect (no causality traveling back in time or anything).
So all in all, to answer your titular question: there is nothing the quantum eraser experiment says about reality that we couldn't already infer from the double slit experiment (essentially just the basic interpretations we know, e.g. Copenhagen). The quantum eraser experiment merely emphasizes how weird the implications are.
I would say the quantum eraser tells us that the question never posed is also never answered. We tend to think of reality as being like the "answer man", where all questions we can think of have answers, we just don't know what they are until we do the experiment. But quantum eraser shows us that the choices we make when we set up the apparatus and take the data actually establishes what happens-- it establishes which questions have answers, and which ones just don't. The delayed choice element shows that the correlations don't exist until they are actualized, there is no "answer man" acting like a fly on the wall of reality.
When the particle is detected before the slits, it abandons wave nature and assumes particle nature on detection and the interference pattern is gone.
When the particle is detected past the slits (very basic delayed choice), it is same process taking place where it abandons the wave nature and assumes particle nature on detection and the interference pattern is gone.
There does not seem to be any difference between the two except the timing. So, it does not seem anything is being erased, or anything is going back in time here.