The paths only seem curved because of the warping of spacetime. Point #1 is actually straightforward to explain: objects simply travel on the straightest possible paths through spacetime, called geodesics. There are actually two different parts of general relativity. 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). They are of course compatible with Newtonian gravity in low-velocity, small-mass regime, so e.g. These equations describe precisely how matter affects space-time.
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). 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). to the precession of the perihelion of the Mercury). for a trampoline you'll get ellipses (well, almost, they do not close completely, leading e.g. You might imagine that if the surface wasn't a sphere but instead was curved differently, the straight lines would also look different. This is one of the effects of the curved space-time on movement on the particles (these are actually tidal forces). 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). 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. 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.
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. I'll try to give you some examples why the straightest line is physically motivated (besides being mathematically exceptional as an extremal curve). Luboš's answer is of course perfectly correct.
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