I know there have been several posts asking the same or similar questions, but I haven't been able to grasp this from reading any existing questions. My question is: why does gravity cause time dilation? To be a bit more specific about what I'm confused about, let me give my current understanding, and hopefully someone can explain to me where I'm wrong.
My current understanding is that although we think of our motion as being through three spatial dimensions, we are also moving through a fourth dimension - time - in a very similar way. Although time appears to be different, it is actually very similar to the spatial dimensions, and together they make up "spacetime". All matter in the universe causes spacetime to "curve", based on how much mass there is. This is why planets orbit the sun - they are really just trying to follow an inertial "straight line", but that straight line is curved due to the curvature of spacetime (this "curved straight line" is called a geodesic). This explains why objects that are already moving appear to "change direction" in the presence of other massive objects - the effect known as gravity.
Additionally, an object that is not already moving, like an apple hanging from a tree, will fall to the earth when its stem breaks off from the tree. This is similar to the reason objects orbit - the apple is "changing directions" to follow the curved spacetime. Although it did not appear to be moving, it actually was - but through the time dimension, not through any of the three spatial dimensions. Because spacetime is curved, the apple does not move straight through the time dimension anymore - it follows a curved geodesic, which curves a bit off the time axis and into the spatial axes.
The more massive the object is, the more curved the geodesic will be - and the more the apple will deviate from the time axis. The component of its motion along the time axis will get smaller, meaning it will "age" less slowly. In other words, due to the curved spacetime, the spatial dimensions are "stealing away" motion from the time dimension. This also explains why when objects move faster, time slows down - they are "stealing" motion in the time dimension to add more motion to the spatial dimensions, so that the four components always add up to the same number.
This is where I feel my understanding falls apart. Before the apple fell, it was held onto the tree by its stem. This was preventing it from moving in the spatial dimensions, and so all of its motion stayed along the time axis. Yet since it is still in the presence of a gravitational field, it experiences time dilation; time moves more slowly, even though the spatial dimensions are not "stealing" motion from the time dimension. Why is this?
The only thing I can think of is that spacetime not only curves, but stretches - so although the apple is still only moving along the time dimension, the axis is stretched out - so it needs to travel "further" to get to the same point, in effect slowing down time.