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Your question is a good one and brings out the problem of using metaphors to explain phenomena.
So the bit of the story that is missing is the statement that all particles should travel in "straight lines" unless acted on by a force. Now on a curved surface we need to understand what we mean by a "straight line". Well a good definition is that a straight line is the shortest distance between two points. In flat space this corresponds to our usual idea of what straight is. But on a curved surface the shortest distance between two points is curved in a very particular way. If you want to get idea of how then look at the flight trajectories of aircraft going from point to point around the earth. (Airlines are very keen to make sure they don't fly further than necessary). So the "straight line" motion on a curved spacetime appears curved from the perspective of someone who thinks the space is flat. That trajectory is the motion of particles in a gravitational field. So in general relativity the perceived curved motion of objects due to gravity is just "straight line" motion (in the sense described above). So curvature=gravity...