I think the picture you have of space being curved is incomplete. In GR, it's spacetime that is curved, not just space.
To visualize GR, you must learn to picture worldlines instead of trajectories. Worldlines are paths of objects through spacetime. The worldlines of freely falling objects are geodesics.
In GR, the presence of mass-energy results in geodesic deviation which roughly means that two initially parallel geodesics will not remain parallel.
So, here's the picture you should have. In flat spacetime, the worldlines of two spatially separated objects that are not moving with respect to each other are parallel.
In the curved spacetime of GR, the geometry is such that the these two worldines converge even if they were "parallel" (not moving with respect to each other) at some point in the past.
Viewed as a trajectory in space rather than a worldline in spacetime, you see two objects falling radially towards one another.