Just to get that out of the way, "straight line" here means geodesic on a curved surface/manifold, but I guess you understand that.
"If the Earth travelling around the Sun is just moving along a straight line in the curved space, shouldn't light also be trapped in orbit around the Sun?"
As you've guessed, they don't follow the same geodesic because of their velocity. And the velocity of a body is automatically taken into account, because you don't just compute these geodesics in curved space, but in curved spacetime, where this makes a difference: Imagine a $t$-$x$-diagram and how the earth or a ray of light go away from the origin. The earths path will be close to the time axis, while the light path will be leaning towards the space axis (depending on your units). Both paths are similarly affected by the curvature of spacetime but they clearly start out in different directions (in spacetime, not in space) and so the geodesics will be quite different. There is some specific angle for which the object will be orbiting (notice that this now requires at least two spatial dimensions). Smaller angles will fall towards the earth while bigger angles will boldly go where no man has gone before. This is a very geometric notion of escape velocity.
"I expect the actual equations in the theory also take into account a body's velocity, not just the curvature."
Since you have 1000 rep on SE Mathematics I can formulate it this way: The geodesic equations are, like Newtons second law, second order and so they require two initial values. One is a position vector and the other one is a velocity vector, i.e. the direction in spacetime. The curvature (and that's another business) is taken into account by the coefficients of $\Gamma$ in the differential equations. These coefficients are basically derivatives of the spacetime metric, which itself must be a solution of the Einstein equations.
Also, if you take a look at the geodesic equation, you see that for $\Gamma = 0$ (flat spacetime), you get the easy case $x''(t)=0$, or $x(t)=x_0+v_0t$, which represent actual straight lines. Here $x_0$ and $v_0$ are initial data.
Furthermore, due to the equivalence principle in general relativity, the geodesics don't depend on the mass of the objects. All things fall equally, once they have the same starting position and velocity. But if two different masses are initially at rest (or since this is a relative statement let's better say: not moving relative to each other), then because of the relation between acceleration and mass, it is more difficult to get the heavier mass to have a certain starting velocity, i.e. direction in spacetime. And therefore you personally will never be able to get a chair on the same trajectory as a pen, which you forcefully throw in a certain direction in space. The chair is too heavy for you to make it follow the same path you could make a small pen follow. If two objects with different masses are initially at rest (not moving relative to each other) and then get pushed equally hard by some force, they will not both end up orbiting the same thing.
So even if the geodesics of spacetime don't depend on the masses of the objects, which would follow them, you'll never see a ray of light follow the same trajectory as a flashlight, because, by the laws of relativity, the flashlight can't move at the speed of light. This is the extreme example: Massive objects never follow light-like geodesics and the other way around.
Imagine you live in a universe governed by extremely simple rules, like Conway's Game of Life, for example. Once you discovered those rules, you might wonder, "Why do cells come alive if they have three living neighbors? Why do they die if they have one? How does that work?" (By "how" here I am referring to "what underlying mechanism makes it work?", which is my interpretation of "how" in the original question.)
In a simulation of the Game of Life that you run on your computer, there is a good answer to this. You can examine the source code, look at the hardware of the computer, and eventually arrive at a complete description of exactly what goes on such that little squares on your computer monitor light up and go off according to the rules.
But we're imagining that these rules are just how the universe works. In that case, there may be no reason at all. Maybe it just does it, full stop.
As humans, though, I think we might find that very hard to accept. There are many cellular automata very similar to the Game of Life, but their behavior is not nearly so rich. Why did we get the one universe with the interesting laws? And how does the universe know to implement those laws without screwing up? Surely there must be some wheels and gears beneath there!
That sort of curiosity is extremely important for physicists, and it has led to a lot of new understanding. Peter Shor pointed out in the comments that wondering about how quantum mechanics works led to quantum information and computation. Famously, Einstein wondered about how electromagnetism worked, leading to understanding relativity. Frequently, a theory of physics doesn't quite feel right to us. That drives curiosity. We demand an answer, and eventually it leads to breakthroughs and new theories.
Physicists have derived great benefit from this approach of taking the pieces that don't feel right or don't feel well-enough explained and using that as a springboard to go deeper, but sometimes it also leads to complete frustration. It turns out that the universe isn't obliged to be the way we want.
If you lived in the Game of Life universe, once you figured out the rule it was following, you could keep asking forever, "Why does it have that rule? How does it implement it?" without getting anywhere. The rule itself is just a short little description. It just says that there's a grid of cells and that they light up and turn off according to a simple pattern, and that's all it says. If there was nothing deeper going on than that, oh well. We wouldn't have to give up trying to find a deeper explanation, but we aren't owed one.
My argument is that real laws of physics are the same. So in General Relativity, we posit that the Einstein equation is true. The theory of General Relativity itself makes no comment on this, just as the theory of the Game of Life makes no comment on why cells with three living neighbors come alive.
So when you ask, "How does something about the mass energy tensor alter geodesics or 4-velocity vectors? I see no explanation of gravity in GR merely a more detailed description of the motions it effects," you are right. GR doesn't say how it does it.
It could be that there's an explanation, but it doesn't seem likely to me that the fundamental problem will go away. For example, suppose someone tells you that gravitation works by sending particles called gravitons around, and gives a detailed description of the theory of gravitons. Couldn't we then ask the same question? How do the gravitons interact with spacetime? We could describe the precise mathematical rules, but fundamentally, this anthropocentric feeling of dissatisfaction would remain. Why those rules for gravitons? If they're derived from some set of appealing principles, why those principles?
Elsewhere in physics, how do wavefunctions know to obey the Schrodinger equation? What forces them to obey that equation rather than doing something else? Nothing. They just do that. It's purely a description of how the wavefunctions behave. The problem is the same, as far as I can see. (You can recast QM in some new formulation, but I don't think this averts the "problem".)
To answer your question as best I understand it, you are right that GR is just a description, nothing more. That may not always be true for GR in particular, but it seems likely to me it will always be true for something. (I can't say for sure, of course, since I don't know what the "something" will be!) It is the nature of theories of physics to be just descriptions. We don't have to accept that as a final word, and our desire to understand more deeply fuels our greatest communal quest for knowledge, but ultimately the universe will do what it will do, and can't be bullied into explaining itself just because things don't feel mechanistic enough for us.
note: This answer is completely rewritten after reading the helpful comments from Qmechanic, Peter Shor, and dmckee. Thank you for your input. This answer is essentially philosophical, so disagreement on it is inevitable, and it represents only my personal opinion.
Best Answer
Yes, that's a fair description of what happens though of course from the ball's perspective it isn't moving - the rest of the universe is moving around it.
However statements like this, while true, give little feel for what's going on. Actually it's extraordinarily difficult to get an intuitive feel for the way spacetime curvature works (or at least I find it so!). The notorious rubber sheet analogy gives a fair description of the effect of spatial curvature, but neglects the curvature in the time coordinate and the time curvature is usually dominant since $dt$ gets multiplied by $c$ in the metric.
The motion of the ball is described by the geodesic equation, but a quick glance at the article I've linked will be enough to persuade you this is not an approach for the non-nerd. I have never seen an intuitive description of how the geodesic equation predicts the motion of a thrown ball.