The diagram is shown below. Each curved line represents the path of a single object through spacetime. Relative to each other, the curved lines are locally parallel, meaning that they represent non-accelerating (i.e. drifting) objects relative to their own local internal spacetime, but these paths are curved relative to an external observer, meaning that they represent acceleration relative to an external observer. If the difference between the local internal spacetime and the externally-observed spacetime is brought about by gravitational spacetime curvature, then this shows how gravitational spacetime curvature brings about acceleration. The problem for most people is that they envision that spacetime curvature is space curvature only and they forget that the warping of time also plays an important role. Acceleration is a change in velocity, and velocity is the change in position over change in time. Therefore, a changing rate at which time flows will necessarily change a velocity value and therefore cause an acceleration relative to an external observer.
An object moving along a straight line at constant speed within curved spacetime relative to its local spacetime looks to an external observer like an object falling and accelerating along a curved path in a non-curved spacetime. It is the spacetime curvature that causes the falling. If you release an apple from at rest and it speeds up toward the ground, fundamentally this is not caused by any force, but by the warpage of spacetime itself. Within the local spacetime and relative to its spacetime, the apple is simply continuing to drift on a straight line at constant speed as is the natural tendency of all objects that are experiencing no forces. However, because spacetime itself is actually warped, to the external observer, this drifting motion through a warped spacetime is falling and accelerating motion. At this point you might say, but the apple was initially at rest and not drifting. However, don't forget that the time dimension is part of spacetime. An object that is motionless in space is still traveling along a path forward in the time dimension.
