General Relativity posits that matter curves spacetime, such that geodesics point towards the object in question, hence, gravity. Now, how does matter do this? What is spacetime "made of", anyway, such that it should interact with matter, being bent by it and forcing it to accelerate (via gravity)?

-
Easy, via gravitrons. The next level we don't understand ... –  Lord_Gestalter Jun 17 at 6:13

The image of space being bent is just an analogy, it is not meant that anything is actually being deformed. Gravity distorts the notion of distance on spacetime, i.e. the presence of matter somehow causes the metric to change. A way to visualize this is to think of spacetime being bent, as you say, but really, spacetime is not made of anything, the idea of an ether has been laid to rest for a hundred years now, with good experimental reasons.

Spacetime interacts with matter since matter exists within (or on, in some terminology) it, and when the notion of distance changes, the behaviour of objects relying on that notion changes.

As for why the presence of matter itself influences the metric...well, this is the defining property of having mass/energy, just as generating (or reacting to) an electric field is the defining property of having an electric charge - in a manner of speaking, mass could be seen as the charge of gravity, though, since we do not fully understand gravity (yet), this is necessarily vague.

-
So... then, how do we separate space from the metric defined upon it? –  silvascientist Jun 17 at 16:17
Short answer: We don't. Slightly longer answer: On the same "spacetime", which is really just what one would call a (smooth) manifold, there are many different possible metric structures one could define. The one that is realized in nature is given by the solution of the Einstein field equations, where the solution is mainly dependent on the energy-momentum tensor, which arises from the distribution of energy/matter. Spacetime without metric is (likely) just $\mathbb{R}^4$. –  ACuriousMind Jun 17 at 16:38

Conventional theory calls it spacetime, which I would assume refers to an unidentified amount of space over an unidentified amount of time but both distance and time are measurements of properties of matter and are dependent upon relative matter, example. Time is the measurement of relative movement, thedistance the sun can travel in a second is different than the distance an atom could due to a difference in the motion of each. Should there be no motion, there can be no "time" as there would be no cycles taking place. 1 earth cycle=1day. Motion=time. In the interest of answering your question with the understanding I have I would have to say space is made of a medium of extreme density, so much so that the substance that comprises it doesn't give any energy (exert a force) as it doesn't require loss of energy to travel through. (when you throw something in space it goes on until acted upon by another force=no drag) there must be some medium by which matter can interact/propagate because it is difficult to believe that spacetime doesn't exist and isn't real or that it's empty yet it's able to react to gravity to influence light...

-