Is the nature of the universe, on a universal scale, fundamentally different from what I see around me?
Yep, that's it.
Spacetime, as it turns out, is a "field", or better yet, an arrangement or "manifold" of "fields". What we call matter and energy, and some other stuff, are actually just "waves" or "disturbances" or "indentations" of these "fields".
I'm putting all these words in quotes in order to emphasize that they don't even mean what they normally mean in spoken language, they are just chosen as the most useful exemplifications of the stuff the math seems to be saying. To be overly dull, the "fields" aren't meadows, the "waves" are very little like water or even sound, even the "particles" aren't little parts at all, at all.
For this same reason, I may have some portion of this explanation "wrong", but I am not overly concerned about that, because even the best explanation that cavediver coughs over here in a bit will only be "more representative", not really truly "right".
We are abusing English, in other words; the proper language is mathematics.