Surely space-time Curvature does not explain gravity, it just describe its effects? In special relativity co-moving objects see the other's 4-velocity as being only temporal.
When they move relative to each other they see the other's 4-velocity has rotated so that it points less in the temporal direction but now has a spatial component.
By the equivalence principle two co-moving objects falling toward a planet see each other's 4-velocity as only temporal in their own (falling) rest frame so they must think the space between them is attached to their rest frame over time. It thus seems that space has the same rotation of its own 4-vector over time (up to a constant if the objects started with a fixed velocity before falling) But surely space does not fall. Also if space-time curvature causes objects to fall, how? I'd have thought it's just a map of how objects move. not a cause of that motion, but if it does cause falling , how? Space isn't moving so as to push or rotate mater. Surely it's curvature it's just a map of the rotations in (light and) matter's 4-vector? 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.
 A: 
so they must think the space between them is attached to their rest frame over time.

You seem to be mixing up spacetime, the dynamical (changing over time) arena in which everything takes place, whose ontological status we can debate endlessly, with coordinates, which are a completely physically meaningless way of assigning numbers to spacetime. Just because two comoving observers agree on coordinate systems doesn't say anything whatsoever about what spacetime is doing. Two planes can be flying in formation, and just because they agree on a convenient set of numbers to describe their position, doesn't mean the air between them is moving with them.

I see no explanation of gravity in GR merely a more detailed description of the motions it effects.

Well, the whole point of GR is that gravity is not a force. Ignoring speculated quantum gravity theories (none of which remotely works at the moment, by the way), gravity is nothing at all like another force. It is just a result of curved spacetime. It is not clear what kind of further explanation you might want. And really, if you fully describe the physics of how everything moves relative to everything else, what more could you want in a theory?
The best I can say as to "how" this occurs is that mass/energy/momentum/etc. alter the distances between neighboring points, and this changes the "natural" path for things to move along. Suppose you distort the surface of the Earth to bring London close to New York - then there will be short "great circles" (i.e. geodesics) that connect them, where there weren't any before.
A: May I suggest that your premise that "Space isn't moving so as to push or rotate matter" is the source of your problem in appreciating the General Relativity explanation of gravity.
The GR perspective is that the very fabric of space is accelerating from outers space towards the centre of the earth.  A freely falling object feels no force. It has not moved from its spot on the fabric, it is merely riding on the space fabric accelerating towards earth.
A person standing on the earth's surface feels the force of the ground pushing up on him.  He is in fact accelerating through space which is 'falling' past him.
The GR perspective therefore eliminates the need for a force called gravity to explain motion.
Of course as to the deeper question why mass affects space in this way, I am sure Science will discover deeper insights but for now I am satisfied with "because thats how it behaves".
A: 
Also if space-time curvature causes objects to fall, how?

Geodesic advance in distorted space-time results in coordinate acceleration towards the mass:
http://www.youtube.com/watch?v=DdC0QN6f3G4

I see no explanation of gravity in GR merely a more detailed description of the motions it effects.

Physics is about describing observed effects quantitatively. If you seek "explanations" and "causes" beyond that, you have to look somewhere else.
A: Curvature affects motion by making the lines that are as straight as possible end up converging, just line how if you and your friends fly at constant altitude from the north pole, then no matter what directions you go (even if you and your friend head out in very different directions) then you start to converge on the south pole. This is a very good way to describe an effect that is determined by the path and not by the mass of the object taking the path. This is sometimes described as "spacetime tells matter how to move" but really this is just that the straightest possible lines converge when spacetime is curved the right way.
How does this relate to those worldlines near a planet?  They try to move in the straightest possible lines, and to first approximation everything is exactly as in SR, so you are right there, but that isn't gravity or curvature, that's just that curved things don't look curved when you zoom in.  So if you were Newton you'd say that things get pulled to the center of the planet so things of a certain size would start to move closer to each other as they each fell towards the same center of the planet.  That's what curvature does, and it does it in a way where the path doesn't depend on whether a light or a heavy particle does it.
But I'll say this, that isn't gravity.  That's just curvature.  And while the commonest effects of gravity are actually the effects of curvature, they are different things.  Let's look first at curvature some more to see what causes it and then we can compare that to gravity very carefully. Something that is usually not mentioned enough is that while mass, energy, momentum, stress, and pressure are sources of curvature, they are not the only things that create curvature, curvature itself can create further and additional curvature. A gravitational wave can propagate or even spread in a vacuum of empty space devoid of all mass, energy, momentum, stress, and pressure.
The region outside a symmetric nonrotating static star is curved, even the parts far from any mass or energy or momentum or stress or pressure. The space remains curved because the existing curvature is exactly shaped so as to persist (or otherwise cause future curvature exactly like itself).
So curvature allows and sometimes requires more and/or future curvature, just as a travelling electromagnetic wave allows and/or even requires there be more electromagnetic waves elsewhere and/or later. The vacuum allows curvature far from gravitational sources just as it allows electromagnetic waves far from electromagnetic sources. What electromagnetic sources allow is for electromagnetic fields to behave differently (namely to gain or lose energy as well as move in different ways and gain and lose momentum and stress). Similarly what gravitational sources do is allow curvature to react differently to itself than it otherwise would.
Imagine a flat region of space shaped like a ball, then imagine a funnel type curved space where two regions of surface area are farther apart than they would be if flat (like a higher dimensional version of a funnel and on a funnel surface two circles of a particular circumference are farther away as measured along the funnel then if two similarly sized circles were in a flat sheet). On its own, spacetime doesn't allow itself to connect those two kinds of regions together, but that mismatch is exactly the kind or not-lining-up that putting some mass or energy right there on the boundary fixes. So without mass those two regions can't line up, with mass they can. Just like an electromagnetic field can have a kink if there is a charge there.
So your curvature likes to propagate a certain way, and if you want it to deviate from that, you need mass, energy, momentum, stress, and/or pressure. And you'd need the right kind to get it to match up, the kind you want might be available, and might not even exist, so not all kinds of curvature will be allowed. But the point of a source is that it changes the balance between nearby curvature and not that affects future curvature. So there is a kind of balance, and there are things that can warp that a balance. Those things that warp that natural vacuum balance are called gravitational sources.
Having curved spacetime is something we observe. Having gravitational sources that can change the normal or usual way curvature evolves is something else entirely. We can make theories about how the sources evolve, and then the curvature is forced to co-evolve with it, and that's what gravity is about, about gravitational interactions (source and curvature together) changing how the curvature evolves changing the evolution that the curvature otherwise would have evolved a different way.
So there is nothing circular, curvature is observed, and on its own it interacts and affects itself in a particular way (that is also observed), but gravitational sources get to change that and by interacting with the gravitational sources (which we can do) we can ourselves make the curvature change in different ways than it otherwise would!
So now let's get to your questions:
If space-time curvature causes objects to fall, how? 
Spacetime curvature (if in vacuum, and sometimes even in some kinds of matter) makes paths that are straight as possible converge.  And that, not falling is what gravity does.  Objects on the surface of the earth are stressed and are pushing to resist that stress and that's why you feel the floor push up on you which is why you don't move inertially, you don't fall.
Is space  moving so as to push or rotate mater?  No, but matter does expand and push when it is under pressure, that;s why you feel the floor pushing up on you.
How does something about the mass energy tensor alter geodesics or 4-velocity vectors? 
It does not alter geodesics.  You move along geodesics because spacetime is curved.  Spacetime is curved differently than it would be all on its own because the sources allow more novel arrangement.
Curvature is one thing, and there is one way it evolves.  When there is are gravitational sources, even more manifold possibilities are available.  Gravity is about gravitational sources, which allows different curvatures.  The curvatures are what is observed.  Keep gravity, and the effect of sources on curvature, separate from the curvature itself which can influence itself.  It's like keeping in mind the difference between an electric charge and an electric field.
A: 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.
A: 
Also if space-time curvature causes objects to fall, how? I'd have thought it's just a map of how objects move. not a cause of that motion, but if it does cause falling , how? Space isn't moving so as to push or rotate mater. Surely it's curvature it's just a map of the rotations in (light and) matter's 4-vector?

Maybe a more "timeless" perspective helps: Try to take the viewpoint of the whole 4-dimensional universe, locally $\mathbb{R}\times\mathbb{R}^3$ at once, such that there is nothing changing. 
Withing this 4-dimensional manifold, the worldline is a unchanging 1-dimensional lin. Only if you want, it can be parameretized alla $s\mapsto  (s,\gamma(s))=p(s)\in \mathbb{R}\times\mathbb{R}^3$. Then $p(s)$ is a point on that line. 
Realize that e.g. the computer monitor you look at in this moment gets identified with the computer monitor you look at 10 seconds later. If you do that, then the computer monitor is an object of infinite (huge at least) temporal length in spacetime and the idea of it moving thorugh time merely emerges from the curiosity that you can only percieve one moment in time at once. 
In the 4-dimensional view, the curvature of all of spacetime never changes (because only 3-dimensional submanifolds can be described as evolving) and hence general relativity, if solved perfectly for all matter in the region it's valid, gives one and only one curvature of spacetime. Then, given that curved manifold the "pushing" really only describes how you are always only in the present and being able to remember the past and to compare it with now, makes you perceive the notion of change.

Apart from that, I agree with Mark Eichenlaub and view physics as descriptive.
A: A model i used to explain it is this.  It is particularly true of non-euclidean geometry.
All space is curved.  Curvature is a measure of circumference per angle.  Straight lines divide the circumference.  A large mass would cause space near it to become more negatively curved, which would create more circumference in the direction towards it.
Gravity can be viewed as a kind of tension in the fabric of space, and moves towards where the circumference is greatest in the angle.  
A: Curved space, or "spacetime" does not explain gravity. It could conceivably explain a curved trajectory of an object, But it has no answer for why a stationary will move towards another object. Without a force of gravity the objects will not move.
