Those depictions offer an analogy for how massive objects warp spacetime that's easier for us to picture than the reality, which is a four-dimensional manifold being squeezed, stretched, and warped by those objects' gravity. A sheet of fabric is two-dimensional, so showing it "stretched" into a third dimension demonstrates gravity's action. You're right in thinking that we would have to be able to think in five-dimensional terms to envision the action of gravity on spacetime in a similar way, since spacetime is four-dimensional.

Humans as a whole probably won't get better at getting intuitive feels for high-dimensional spaces and manifolds. 3D is "baked in" to our perception of the world and our internal model of how it works and how to move through it. Mathematicians often do get slightly more of an intuitive sense from working with them long enough.