Why does mass naturally move closer toward's the center of other masses? If general relativity is understood to mean that mass curves space such that objects are still traveling in a straight line, why does mass naturally move closer toward's the center of other masses?
Think at the surface of a ball. A straight line within this curved surface is a great circle.
Take some point on the ball. Draw two great circles (apparent straight lines in the surface) starting in this point, with some non-zero initial angle. The two apparent straight lines intersect at the opposite side of the ball.
Wrongly assuming the surface of the ball being flat would result in the conclusion, that the lines must be curved.

The same applies to our three-dimensional space. We intutively assume it as flat, such that two lines starting at an observer with some angle should never intersect elsewhere. But in some cases they do; so we intuitively think the lines must be curved. But according to general relativity, it's actually our space-time which is curved by masses.

As an example, a ballistic curve looking like a parabola is actually a straight line in the curved space-time. So, two objects moving in different directions in e.g. Earth's orbit can eventually hit each other despite lack of drag or propulsion. They both move on a straight line (geodesic) in a curved space-time. The curved space-time allows these to lines to intersect later. Our three-dimensional Euclidean intuition interpretes this as curved trajectories.