How exactly does curved space-time describe the force of gravity?
Luboš's answer is of course perfectly correct. I'll try to give you some examples why the straightest line is physically motivated (besides being mathematically exceptional as an extremal curve).
Image a 2-sphere (a surface of a ball). If an ant lives there and he just walks straight, it should be obvious that he'll come back where he came from with his trajectory being a circle. Imagine a second ant and suppose he'll start to walk from the same point as the first ant and at the same speed but into a different direction. He'll also produce circle and the two circles will cross at two points (you can imagine those circles as meridians and the crossing points as a north resp. south poles).
Now, from the ants' perspective who aren't aware that they are living in a curved space, this will seem that there is a force between them because their distance will be changing in time non-linearly (think about those meridians again). This is one of the effects of the curved space-time on movement on the particles (these are actually tidal forces). You might imagine that if the surface wasn't a sphere but instead was curved differently, the straight lines would also look different. E.g. for a trampoline you'll get ellipses (well, almost, they do not close completely, leading e.g. to the precession of the perihelion of the Mercury).
So much for the explanation of how curved space-time (discussion above was just about space; if you introduce special relativity into the picture, you'll get also new effects of mixing of space and time as usual). But how does the space-time know it should be curved in the first place? Well, it's because it obeys Einstein's equations (why does it obey these equations is a separate question though). These equations describe precisely how matter affects space-time. They are of course compatible with Newtonian gravity in low-velocity, small-mass regime, so e.g. for a Sun you'll obtain that trampoline curvature and the planets (which will also produce little dents, catching moons, for example; but forget about those for a moment because they are not that important for the movement of the planet around the Sun) will follow straight lines, moving in ellipses (again, almost ellipses).
There are actually two different parts of general relativity. They're often stated as
- Spacetime tells matter how to move
- Matter tells spacetime how to curve
Point #1 is actually straightforward to explain: objects simply travel on the straightest possible paths through spacetime, called geodesics. The paths only seem curved because of the warping of spacetime. If you're a physicist, then you would want to know that that fact can be derived from the principle of extremal action (with all the requisite mathematical details), but if you don't want to wade through the math, hopefully it at least makes sense that objects move on "straight" lines. There is no actual force involved when a massive (or even a massless) object's trajectory curves in response to gravity, because it doesn't take any force to keep something moving on a straight line. (I can definitely expand on this point if you want)
Now, I mentioned that spacetime needs to be warped in order for objects' trajectories to appear curved to us despite them actually being "straight." So the essence of point #2 is, why is spacetime warped in the first place? Physics doesn't have a good answer to that. Technically, we don't have an answer to point #1 either, but the "straight line" argument at least makes it seem plausible; unfortunately, there's no equivalent plausibility argument for why spacetime warps itself around matter. (Perhaps someday we will come up with one) All we can do right now is produce equations that describe how spacetime behaves around matter, namely the Einstein equations which can be written $G_{\mu\nu} = 8\pi T_{\mu\nu}$ among other ways.
the trampoline analogy needs an extra source of gravity - because this is what the laymen, the recipients of the explanation, intuitively understand - but the real general relativity doesn't need any extra "external" gravity.
Instead, general relativity says that the space is getting curved by Einstein's equations, $$G=T$$ where the left-hand side are numbers describing the curvature at a given point and the right hand side is the density of matter and momentum. I omit indices and constants haha. So general relativity says how the spacetime is curved under the influence of matter.
The second part of the story is that general relativity also says how matter moves in external geometry. It moves along "geodesics", lines that are as straight as you can get. $$\delta S_{action\,ie\,proper\,length} = 0$$ This actually means that the objects move along the predicted, seemingly curved trajectories. These trajectories are actually as straight in the curved spacetime as you can get.
Imagine that there is a hemisphere replacing a disk in the trampoline. So there exists a (nearly) straight line on the hemisphere - namely the equator near the junction with the rest of the trampoline. Note that the equator on the Earth is a maximum circle - so it is one of the straightest lines you can draw on the surface of Earth. The same is true for all actual trajectories that objects choose in spacetime of general relativity.
So in the hemisphere-above-trampoline example, particles can orbit around the equator of the attached hemisphere, just like planets, because it is the straightest and most natural line they can choose. I don't use any external gravity to explain the real gravity; instead, I use the principle that particles choose the most natural - the straightest - line they can find in the curved spacetime.
Best wishes Lubos