Physicist: The original post is here.
The curvature of space alone has almost no effect on the movement of objects until they are moving really fast. With the exception of only the most extreme cases (black holes), space is very, very close to flat. For example, the total stretching of space due to the Earth amounts to less than 1cm. The precession of Mercury’s orbit is another example of the tiny effect of the curvature of space (and it is tiny). Literally, there’s a little more space near the Sun than there “should” be, and as a result the direction in which Mercury’s orbit is elliptical moves. It takes a little over 3 million years for it to go full circle.
In almost all cases the vast majority of an object’s movement is tied up in its forward movement through time. The curvature of spacetime (not just space) is responsible for gravity. Literally, near heavy objects, the “future direction” points slightly down. So anything that moves forward in time will find its trajectory pointing down slightly. This takes the form of downward acceleration. This acceleration (time pointing slightly down) is entirely responsible for the motion of the planets, and every other everyday experience of gravity.
It may seem a little confusing that, once you’re moving, the explanation doesn’t change and falling is still caused by movement through time. Well, there is some effect caused by spacial movement and the spacial part of the curvature, but these effects are almost completely overwhelmed by the effects from the time component of the velocity (much, much bigger).