Why does gravity cause time dilation? 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.
 A: I am afraid your allegoric figuring out why gravity causes time dilation is further confusing. The physical reality is measured by the metric of a reference frame which composes the coordinates, but the single coordinates do not have necessarily a specific meaning in GR (general relativity).  
The Einstein equivalence principle allows to describe gravity in terms of geometry (metric) of a curved spacetime.  
If we consider a static (Schwarzschild) spherical mass, radially the curvature is different and the proper time of stationary observers, as given by the metric, is progressively slowing as you approach the mass, if compared to the time measured by an observer far away from the mass.  
The time dilation of SR (special relativity) is still a different concept as it is symmetrical between two observers in uniform relative motion. Instead the gravitational time dilation is not symmetrical; in fact the stationary observer measures a time contraction if compared to the far away observer.  
A: Jahlon, 
I understand your confusion. Think of your apple hanging on the tree. The four velocity vector IS tipped into the spatial dimension, this component is precisely the force the branch feels pulling the apple downwards.
It might help to consider the related four momentum components and realize that the spatial momentum component of your apple is continually pulling on the branch even though it is not moving in that direction. 
For a horrible analogy, say we have a small toy car whose wheels always rotate at a constant speed across the floor. If we put it at against a wall at some very small angle (lets say a friction-less wall), the car will now go slower along the wall than if it was traveling strictly parallel to it. because now it is also exerting a force against the wall just as the apple is on the tree branch. 
Not sure if I made that clear for you (: Hope it helps!
A: Being in a gravitational field strength $g$ is equivalent to having an acceleration of $g$. Since this implies the body cannot remain at rest in any fixed reference frame, time dilation is to be expected.
Why does being a distance $r$ from a point mass $M$ multiply the passage of time by $\sqrt{1-\frac{2GM}{rc^2}}$? Because that's the value of $\sqrt{1-\frac{v^2}{c^2}}$ when $v=\sqrt{\frac{2GM}{r}}$, the Newtonian-limit escape velocity. In other words, gravitational time dilation is equal to the escape-velocity time dilation in far space, where the weak-field limit is obtained.
A: You seem to be reaching for a heuristic. (Here I mean by 'heuristic' a mental picture that allows you to organize your thoughts.)
If so, I suggest you look into the River model of General relativity
As emphasized by Hamilton and Lisle (the authors of that article), the river model is not a physical theory. It is a tool that helps you see a coherency that you would otherwise not see.

General remarks about physics understanding:
It is always very gratifying when a physical explanation moves our understanding to a deeper level. Example: when you add salt to a container filled with ice (and some melt water) why does that salt/water/ice mix drop in temperature? With the concept of atoms and statistical mechanics you can explain that. It is ever so tempting to expect that all physics theories are doing that for us. 
But there are many cases where you have to just assume something, without explanation, in order to frame a theory at all. The predecessor to GR is an example: Newton's law of universal gravity. In Newton's time there was no prospect of providing an explanation of how gravity could possibly operate over the vast distances of the solar system. Newton argued: the merit of this law of gravity is that it unifies: it describes both the orbits of planets and the gravity on Earth. The physics community accepted the law of universal gravity on that merit.
General relativity is a set of assumptions that together constrain the equations for gravity and motion. But those assumptions themselves are not explained. Einstein chose to state one particular assumption as key: the Principle of Equivalence. You have to assume that, otherwise you cannot even begin to formulate a theory.
The Principle of Equivalence seems to be universally valid. But at present we don't have an deeper theory that offers an explanation as two why the Principle of Equivalence applies in our Universe. The concept of curvature of spacetime arises naturally in the context of the PoE, but it is an interpretation and it's important to not overthink it. I rather suspect that your metaphor of "stealing away" motion from one dimension to another is such over-interpreting of the concept of spacetime curvature.
A: I think it's quite intuitive to explain gravitational time dilation by means of the equivalence principle and a light clock. The EP says you can't distinguish acceleration in flat space from hovering in constant distance to the center of a mass. 
Think of a box with constant acceleration in flat space-time. Light pulses sent from the top are received at the bottom. While the light is still travelling the bottom accelerates towards it. Therefor the clock at the bottom measures shorter intervals between two pulses than the clock at the top. If the top clock shows  say 2 hour elapsed time the bottom clock could show 1 hours (depending on the acceleration). As this scenario is equivalent to the box hovering close to a black hole the bottom clock runs the same amount slower than the top clock if the acceleration is the same.   
A: Because time is a calculated result of distance/velocity. Even though you have a subjective experience of time as a real thing, Time is not an independently existing entity like sound, light or other things you experience.As your velocity increases, people in a stationary rest frame (relative to you) see that distances between things in your frame of reference change. Remember the example of the train traveling near light speed that passes you appears shortened. The stationary people (even tho they might not be in the train station)  therefore calculate that time is moving more slowly for you than it is for them.
A: I do not think there are theories on "Why". But there can be ways to settle things in your mind. I would try this - 
Time is basically a tick rate of events. Suppose breathing is an event. You are at sea level, you have your own rate of breathing. Then you dive 10 feet under water with breathing aparatus. Your breathing rate changes due to the stress/pressure caused by depth. I guess it is harder to breath under water and the breathing slows down. You dive deeper and breathing slows down further.
Same way, existence inside a gravitational field experiences some stress that slows down tick rate for all events including subatomic events. That is why, the gravitational time dilation depends only on gravitational depth, not on gravitational acceleration. So, even in less curved geodesic, time can be slow depending upon the gravitational potential. For example, at center of the earth, there is no geodesic curve (or slope) but time dilation is actually more there as compared to that on surface. So, it is the depth, not the slope of the curve.
Same way, speed causes a stress and slows down events.
There is a relation between the two - gravitational time dilation, and speed time dilation - 
Time dilation in a gravitational field is equal to time dilation in far space, due to a speed that is needed to escape that gravitational field. 
Here is the proof - https://en.wikipedia.org/wiki/Gravitational_time_dilation#Important_features_of_gravitational_time_dilation
