General relativity tells us that gravity is the curvature of space-time, rather than just the curvature of space.
Imagine an object in zero gravity (ie a non-curved space-time). If the object does not have any outside forces acting upon it, the object will not move in space, but the object is still moving in space-time because it cannot help moving in time.
In a curved space-time, when an object with no outside forces acting upon it (ie an object in free-fall) moves through time, it is also moving through space.