Q: Why does “curved space-time” cause gravity?

Physicist: In a flat space local ideas about “parallel” and “perpendicular” are global.  That is, if two lines are parallel, and you follow them for a while, then they’ll still be parallel.  (By “flat” here I mean exactly this property, parallel is parallel forever.  Not just “flat like paper”.  So you can have 2-d flat space, 3-d, 4-d, whatevs)

An example of curved 2-d space is the surface of a ball (just the surface, don’t worry about the inside and outside).  If you draw two parallel lines on the ball, then eventually they will cross.  The curvature forces the two lines to come together.

A straight line on a sphere always traces out a "great circle", like the equator. These lines are parallel twice, but also intersect twice.

If an object is not experiencing any force, then it will travel in a straight line through space.  This is true for space-time as well.  So if you’re sitting still (traveling forward in time), and no one applies a force to you, you’ll continue to sit still (travel forward in time).

Now imagine two people hovering above opposite sides of the Earth.  I say hovering in place because this means the lines they trace out in space-time are (initially) parallel.  As you run time forward you’ll notice that, even though no force is acting on them (don’t say gravity) and they are traveling in straight lines through space-time, they still move together (fall toward the Earth).

This is due entirely to the Earth curving space-time around it.  Literally, it takes the original “flatness” of empty space, and curves it.  It’s a little more complicated because the time dimension and the spacial dimensions are fundamentally different, but not as much as you’d think.

Another, slicker-sounding way to describe gravity is: “Things fall because time points a little bit down”.  That’s not creative prose, I mean that literally.