Cyliverse
The "cyliverse" is a 1-dimensional universe comprised of a circle that evolves over time as a cylinder. Objects within this universe are points that can move back and forth along the circle. An object at rest has a worldline that just runs straight up the cylinder. Two objects at rest near each other form two parallel worldlines running up the cylinder. Hopefully this is very clear in your mind.
Curvature
Now, imagine that the cyliverse has a "dome" at some point. This dome is a half-sphere with the same radius as the cyliverse. Now, this dome does not describe all objects in the cyliverse. Rather, it only describes inertial objects. That is, objects that are not accelerating. What you notice is that if objects are just coasting, when they reach the dome, they all start coming together, as if the universe is heading for a Big Crunch. This is also equivalent to all objects falling into a black hole with no discernable center or location.
Any two straight lines you draw on this dome will meet at the pole. That corresponds to worldlines in which any two objects at rest will end up moving towards each other for no apparent reason. Note that we have not invoked any explicit forces to cause this motion. We did not conjure a rocket or asteroid or laser. All we did was curve spacetime. And all the objects have to do is not try to move. And yet, by sitting still, they end up moving anyway. That's because spacetime itself is curved in such a way that forces this motion.
This is a very crude illustration that is a little bit misleading, but it should still give you a sense of what it means for curved spacetime to cause gravitational motion.
A Star is Born
A more realistic picture would be to imagine us manifesting a large mass somewhere on the cylinder. If we look at the ring of the cyliverse as a clock, we can say that there is now a large mass at 3 o'clock. What does that do to the worldlines near it? Well, instead of remaining straight, it tilts them all towards 3 o'clock. And the closer the line is to 3 o'clock, the more it tilts. Lines far away (like at 9 o'clock) are curved so little you can't even tell the difference. We could say that the curvature mostly disappears around 1 and 5 o'clock. What does it mean for the worldlines to be curved? It means that an object at rest sitting at 2 o'clock will move towards 3 o'clock. It's like someone took all the worldlines around 3 o'clock and pinched them together in the future. But they didn't just pinch them at one spot. They pinched them at every spot, but for worldlines that started a particular distance in the past.
So if you are at rest at 2 o'clock, it might take 1000 s to fall into the star at 3 o'clock. That means that your worldline starting at 2 o'clock at $t_0$ curves towards 3 o'clock, intersecting with it at $t_0+1000$. But there is another worldline at 2 o'clock, $t_0+8500$, and it curves into 3 o'clock at $t_0+9500$, etc. So the cyliverse is warped in this way, where all the worldlines around 3 o'clock curve into it no matter what time they start.
In order to not fall into the star, you have to get off the geodesic by applying a force away from it. You can "hover" a fixed distance from the star by applying thrusters which exactly counteract the curvature of your worldline. But you must expend a force to do so, because the inertial path will bring you towards the star.
Conclusion
The curvature of space causes motion for inertial objects by decreasing their relative distance in the future. Gravity curves space by causing inertial worldlines to bend towards the gravitational source. And thus, inertial objects on these worldlines will naturally come closer together, because that is what their geodesics do. A train follows the track around a curve because that is where the track leads it. Objects in a gravitational field also "follow the track", and are "pushed along" by time.