The other answers are more or less correct, but pehaps I can say something more to the point of the question, «¿how is curved space-time itself actually capable of exerting a force? ».
No force whatsoever is needed.
Gravity is not a force. What is a force, anyway? Newton clarified for almost the first time in Science what a force is: First I will say it, then explain it: A force is something which makes the motion of a body deviate from uniform straightline motion.
Newton pointed out that bodies have a tendency, inertia, to continue in whatever direction they are already going, with whatever velocity they have at the moment. That means uniform, rectilineal motion: steady velocity, same direction. Newton actually knew this was what would be later called a geodesic, since « a straight line is the shortest distance between two points ».
Newton then went on to say that to overcome inertia, to overcome this tendency, requires a force: force is what makes a body depart from the geodesic it is (even momentarily) headed on (its direction and speed).
It was then Einstein (and partly Mach before him) who said this does not get to the essence of the question. For Einstein, any coordinate system had to be equally allowable, and in fact, space-time is curved (as already explained by other posters). A body or particle under the influence of gravity actually does travel in a geodesic....i.e., it does what a free particle does. I.e., it does what a particle not under the influence of any force does. So gravity is not a force.
Newton did not realise that space-time could be curved and that then the geodesics would not appear to our sight to be straight lines when projected into space alone. That ellipse you see in pictures of planetary orbits? It is not really there of course since the planet only reaches different points of the ellipse at different times...that ellipse is not what the planet really traverses in space-time, it is the projection of the path of the planet onto a slice of space, it is really only the shadow of the true path of the planet, and seems much more curved than the true path really is.
( ¡ The curvature of space-time in the neighbourhood of the earth is really very small ! The path of the earth in space-time would even appear to be nearly straight to an imaginary Euclidean observer who, in a flat five-dimensional space larger than ours, was looking down on us in our slightly curved four dimensional space-time embedded in their world. It's $ct$, remember, so the curving around the ellipse gets distributed over an entire light-year, and appears to be nearly straight...and is straight when one takes into account the slight curvature of space-time.)
Since every particle under the influence of gravity alone moves in a geodesic, it does not experience any force that would make it depart from its inertia and make it depart from this geodesic. So gravity is not a force, but electric forces still do exist. They could overcome the inertia of a charged body and make it deviate from the geodesic it is headed on: change its speed and direction (when speed and direction are measured in curved space-time).
Einstein (and me too) did not want to change the definition of force in this new situation, since after all electric forces are known to exist and are still forces in GR. So the old notion of force still retains its usefulness for things other than gravity. To repeat: if a body is not moving in a geodesic in space-time, you go looking for a force that is overcoming its inertia....but since gravity and curvature of space-time do not make a body depart from a geodesic, neither of them is a force.
See also http://www.einstein-online.info/elementary/generalRT/GeomGravity
which avoids the trampoline fallacy and has a great image of the great circle.