I've been wracking my brain trying to understand what "curved spacetime" really is, and I think replacing one dimension with the time dimension then drawing the world-lines through time was the "aha!" moment.
And this obviously goes ahead and explains our relative perception of gravity appearing to be a constant force downwards, whereas we really just following a curved world line through time (which we are always moving along) that is curved downward toward the Earth.
I added a satellite orbiting to show another aspect, that is that the space station could be said to be constantly moving "up" away from the center of mass, but the curvature keeps it at the same distance to the Earth. We can't see that extra dimension so we feel like we are "falling".
Is this diagram of the world-lines through time correct? The orbiting space station is just an extra thought experiment I put in.
(click image for full size)