Ultimately a physical theory is an assertion that if you go out into the real world and measure certain numbers, then some specific mathematical relationships between those numbers will hold. Imagery like "curved spacetime" is just words to help us intuit the behavior of the equations and can't be proved. In fact Steven Weinberg in his GR textbook deemphasizes the idea that general relativity is about "curved spacetime," writing
This seems off base. Mathematical relationships can often be expressed in many different forms, some amazingly different in appearance. The important thing, again, is the predictions that come out of the math, which must be the same if the different forms of the math are really equivalent.
To give an example that has gotten a lot of interest recently, the "AdS/CFT correspondence" is the statement that two very different specific mathematical theories actually (and very surprisingly) describe the same physics. But these two theories seem very different--for example, they posit different numbers of spacetime dimensions! So one theory might say that spacetime has 4 dimensions while the other says that spacetime has 5 dimensions. Since both predict the exact same results, it becomes clear that the mathematics of a physical theory doesn't tell you how you should picture the universe described by the theory. You can think of it as a 4-dimensional spacetime with one set of physical laws. You can think of it as a 5-dimensional spacetime with a different set of physical laws, and this turns out to be completely equivalent to the other description. You can think of it as a simulation proceeding in a cellular automaton computed by a sentient frog (though this is probably a less useful picture), so long as you posit that the simulation is programmed in a way that corresponds to the predictions of the theory.