I've come up with my own 'thought model' because the rubber sheet thing is pretty much 2 dimensional. If you think of a globe of the Earth, the shortest distance from one side to the other is a 'straight' line along the surface. If you consider it to be made of something like soft rubber, then a weight of some type would make a dent in the surface, which would deflect the course of another object that passes it on the way to the other side. The real shortest distance would be if you poke a stick through the thing.
Space-time is somewhat similar, although far more complex. What to us is a straight line is not the shortest distance between 2 points. The 'stick' through the fabric of space-time, which Einstein referred to as a 'geodesic', is the actual shortest path. That's one reason that some of us hold out hope for possibe 'faster-than-light' travel or communications, if we can somehow access the geodisics rather than be forever restricted to normal space. (This is a very scant hope, because it would appear that the energy requirements would be prohibitive.)
Something to keep in mind as well is that Newton's theory was not so much wrong as it was incomplete, just as Einstein's is in light of quantum mechanics. It still quite adequately expresses anything that you're likely to encounter in normal life, such as how long it will take a dropped bowling ball to land on your foot, or how much it will hurt when it gets there.