How does the curvature of spacetime induce gravitational attraction? I don't know how to ask this more clearly than in the title.
 A: The simple answer is that the curvature of spacetime does not induce gravitational attraction. It describes it.
In all of classical physics (not only general relativity), the motion of bodies can be described as motion along the geodesics of a suitable differential geometry. Body A acts on body B by influencing the curvature of the geodesic along which body B moves. See the illustrations on this page of my personal website for both classical electrodynamics and general relativity.
A: The best way to see this, I think, is using Einstein's 1907 approximation. Einstein noted that gravity slows down clocks, so that where the potential is more negative, clocks run slow, and where the potential is small, clocks run fast, so that the rate at which a clock ticks at position $x$ is $\sqrt{(1-2\phi(x))}$.
Assume that space is completely flat, and only the clock rate changes from point to point. This is a curved space-time, but the space-part is flat. Then you ask: "what is the curve in space-time which is extremal for the relativistic distance from point A to point B?". This is the analog of the question "what is the shortest path from A to B" in geometry.
The integral for this is:
$\int \sqrt{(1-2\phi(x)) - |v|^2)} dt = \int -\phi(x(t)) + {v^2\over 2} $
Which, if you multiply by the mass m, gives you the action for a Newtonian particle in a gravitational potential. So Einstein's law reduces to Newton's. The remaining work is to see that the equation for the time-time component of the metric reduces to Laplace's equation for static masses, and this is not too hard to do, but annoying to do twice or to write up. That this reduction works was one of Einstein's criteria for a good field equation.
A: I'm a bit worried about getting a reputation for citing myself too much, but I'll go for it anyway. (In my defense, I always admit it when I'm doing it!) 
John Baez's and my pedagogical paper The Meaning of Einstein's Equation aims to address exactly this question. We describe the meaning of spacetime curvature and the way that Einstein's equation connects spacetime curvature to the matter content of a region of space. As one example, we use this description to heuristically "derive" Newtonian gravity.
I think the most important point is that, in "ordinary" situations involving particles moving at speeds much less than $c$, the "time" part of spacetime curvature is by far the most important part. Intuition about curved space (as opposed to spacetime) only gets you so far. 
A: If You are in flat spacetime and move from A to B, the easiest way is to travel by fragment of an straight line between points A and B. When there is gravitation, spacetime is bending. So the easiest way is to follow the line which is the shortest, but it is not straight line, because of the curvature. So another choice is the line which is shortest but remain within spacetime, and is "minimal". And this is exactly physical trajectory of small particle in such curved space.
As gravitation mass of a body and inertial mass of a body are exactly equal, there is equivalence between such geometrical description (when curvature is dependent of gravitational mass) with dynamical one which uses forces to describe particle movement ( because inertial mass is equal to gravitational one - so the two pictures are perfectly equivalent). 
So when You go from point A towards point B you follow no stright line but more complicated curve such as parabola or ellipse -  because it is has "minimal feature" among accessible trajectories. Of course You may run some kind of engine, and run on different - non minimal trajectory - for example straight line. This cost You some bill for fuel, and is not minimal at all...
As planets, meteorites, comets, thrown rocks etc. do not have its own engines, they have to move along "minimal" trajectories.. As "minimal" lines in curved spacetime are exactly trajectories of body within gravitational fields, this two pictures describes the same move with different language. Of course attraction has its causes in curvature when You use geometrical language - many trajectories goes close and close each other. So You may say - gravitation is represented by curvature.
A: It is a geometrical theory, so best explained with graphics:
http://www.youtube.com/watch?v=DdC0QN6f3G4
http://www.physics.ucla.edu/demoweb/demomanual/modern_physics/principal_of_equivalence_and_general_relativity/curved_spacetime.html
