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)?
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.
Due to gravitational time dilation, time moves slower in areas of space near heavy objects. You can think of it as time having different speeds at different locations in space. The combination of a three dimensional space coordinate with the speed of time at that location is space-time.
If you define space as the distance between two points, you can create more space by moving these two points farther apart. But this requires energy. How much energy depends on the mass of the two points and their distance. If the two points move closer together,energy is released in the form of acceleration and motion. On the surface of the Earth, increasing the space/distance between a one pound object and the Earth by 3 feet requires roughly 1 calorie (3 foot pounds) of energy.
How does matter curve space like this? If you define space as the distance between two points, then the universe is using polar coordinates, not rectangular coordinates and gravity doesn't exist, but the result of our incorrect idea of what a straight line is. You can think of objects in motion following a straight line and being affected by gravity or objects in motion follow a path curved because space-time is curved. Is the moon following a path around the Earth curved by gravity or is the moon moving in a "straight line" around the Earth?
How to visualize this? If you drew a bunch of points on a piece of paper and drew a line between every point and every other point, places where lots of points were close together would represent a heavy object. Space (the distance between two points) would not be distributed evenly on the page and space would be "thicker" near a heavy object with lots of points.
You can also combine the two equations:
E = MV2 (Energy = Mass times Velocity squared)
E = MC2 (Energy = Mass times the speed of light squared)
to get the total energy of an object including it's motion:
E = MV2 + MC2
This is a simplistic, probably slightly incorrect version of Einsteins 10 equations of space-time. If an object moved from a region of space where time was fast to a region where time went slower such as closer to a planet, C (speed of light) would decrease (due to time dilation) and V (velocity) would increase (object starts moving faster). This answers the question of when an object moves away from the Earth and it slows down, where the energy went.
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...
Space is the canvas on which the Universe is painted. Space is created when a Higgs wave (dark energy) intersects dark matter. When this happens, three things are created. Matter, energy and a whole lot of SPACE. Space has two known properties. First, Dr. E. tells us it can be warped. Second, I tell you it cannot be compressed. That being said, this reaction is still occurring and therefore the Universe continues to expand at an accelerated rate. This also explains the Big Bang. Prior to it the Universe had no dimension as the two darks were both waves and did not require space to exist. A quantum foundation caused the dark energy to produce the cascading Higgs wave which produced the matter energy and SPACE.
Hope that clears things up.
L. Stewart Hearl