Remember that spacetime = space + time dimensions. The object is always travelling through time, and the curvature of spacetime is converting some of that speed through time into speed through space. That's what you perceive as motion (caused by gravity).

Time and space are linked together. The faster you go through space, the "slower" you go through time (as in you experience it slower). This is very measurable and even used to alter timings for satellites GPS readings. You can take an atomic clock on a plane and age slower than someone who just stayed on the ground.

So the spacetime curvature is continuously converting some of your temporal motion into spatial motion, until that's stopped by the surface of the Earth which is constantly "accelerating" to stop you from going further.

As to why we always move through time, that's beyond my understanding at this point but it's a fundamental axiom of physics.