Now just add massive scale and distances.
If you let your current momentum be your direction of facing, and let the same momentum also specify your direction of motion, the Christoffel symbol tells you what your momentum vector would be after an infinitesimal amount of motion. This can be integrated to find the version of a straight line appropriate for a curved surface (imagine an ant walking straight forwards on the surface of a cone or something), a geodesic. A changing momentum is like a force is acting, so that's gravity.
There is more to learn than that, of course. Many many many books have been written about general relativity and you can read them.
The point is that mass bends space-time. The amount of bending is dependent on the size of the mass and on the distance from the mass. Even though the Sun is incomparably heavier than the Earth, it is also MUCH farther away from you. So, space-time around the Earth is curved much more towards the center of the Earth than it is towards the center of the Sun. In the mattress analogy, consider a large mattress, with a bowling ball and a car sitting on it. The car will obviously bend the mattress much more, but if you're close to the bowling ball, you'll still fall towards the bowling ball first before both you and it fall towards the car.
So, say you're in an airplane moving directly forward, with the Sun just overhead (and the Earth obviously just below you). The Earth curves spacetime towards it a lot in this area, while the Sun curves it towards itself just a little bit. The overall curvature is such that time still moves more for the bottom of the plane (closer to the Earth) than the top of the plane (closer to the Sun). So, the bottom side moves a little slower than the top side, but the structural integrity of the plane pulls the top side towards the bottom, causing a slight motion towards the Earth - gravity [note that the GP's explanation got the signs a little wrong - time flows slower, not faster, closer to a big mass]. Conversely, if the Earth disappears from the picture and only the Sun remains, now the top part of the plane will move slightly slower, pulling the bottom part towards it, and thus towards the Sun.
Other than that, thank you for a very clear explanation.
There is no way to have a “zero speed orbit”. You’d be on a trajectory straight in to the middle of the sun or away from it (under your own power). The only way to stop is to push away with equal constant acceleration (which looks like “force”). This is what rockets do.
I always hated the ball and sheet example simply because it was describing gravity with gravity. It felt fundamentally wrong.
Imagine spacetime as a field of local clocks. Far from the Sun, clocks tick faster. Near the Sun, clocks tick slower. A freely moving object tries to follow the straightest possible path through spacetime. But because the “time axis” changes from place to place, what counts as “straight ahead into the future” tilts slightly inward near the Sun. So the Earth’s path through spacetime curves toward the Sun.
Earth’s spatial speed around the Sun is about 30 km/s. But through spacetime, its “timeward” motion is basically c, 300,000 km/s. So even a tiny tilt in the time direction creates a significant spatial acceleration. That is why the time-warping term dominates for slow massive bodies.
Near Earth’s surface, clocks lower down tick very slightly slower than clocks higher up. The change in tick rate is on the order of 10^(-16) per meter. While extremely small, that's enough to generate the familiar 9.8 m/s^2 spatial acceleration we experience. Such a small gradient in clock rates generates macrosopically noticeable spatial accelerations because the "translation" factor is c^2, a tremendously large number.
Now, if I wanted to cover all my bases here, I'd need to point out that gravity does also bend space -- that is just not a relevant factor for "ordinary" gravity acting on relatively slow moving matter (like the Earth itself, or the Earth's atmosphere). For instance, for light itself, spatial bending is just as important (in fact, the gravitational deflection of light by a weak static gravitational field is controlled by a near 50/50 split between spatial and temporal effects). Near a massive black hole, it's not that simple and can't meaningfully be understood in terms of "time" and "space" effects being independently separated.
Edit: The response below is dead for some reason, please vouch.
But this just raises the question of what it means to be larger in time than space. You can look at it in terms of multiples of Planck distance or time, but I think there's a more enlightening way to look at it. If you express the speed of light in those Planck units, it's 1. But the speed of light is also the maximum speed of causality. Any causally-bound system must run long enough for chains of causation to propagate, usually far below the speed of light in practice. This means that basically anything that exists within the bounds of our manipulation must be happening at scales where there is far more time involved than space.
We all exist below the diagonal because the diagonal is the bound at which the ways chemistry and biology work no longer even are theoretically possible.
It may help your intuition to consider the extreme case of a black hole. The event horizon is where time is so warped that no possible future trajectories lead outside of the black hole, and you need a magical time machine to escape. (Of course, the best way to gain intuition is to work through the mathematics, either symbolically or with diagrams, rather than reading English-language descriptions.)
There is a sense in which an orbit is a straight line. Obviously, an orbit is not a straight line through space (unless you count the perfect and unobtainable orbit of a beam of light around a black hole, some distance from the event horizon), but we often think of them as spirals through spacetime: there's an argument that really we should think of them as straight lines through spacetime, much like how a great circle is a straight line along the earth's surface.
https://www.youtube.com/watch?v=8yhk1EZq9tY
fortunately that video is more gentle but the math in that youtube channel absolutely melts my brain some days, I can keep up for the first minute but then all bets are off as he dives in and I realize there are some insanely brilliant people out there