Is this explanation to laymen of how curved space time makes you experience gravity correct? I always found it hard to explain to the lay person how the gravity they are feeling is caused by curved spacetime. Their reaction was just one of bafflement and they ended up thinking that it is just too advanced to really understand.
There is an excellent answer to how curved space time causes gravity here: How does "curved space" explain gravitational attraction?, but that level of explanation is exactly of the mathy kind that would make the layman's eyes glaze over.
A few years ago I came up with an analogy which made intuitive sense to me and people were starting to "get" when I explained it to them. I don't have a deep background in the math of general relativity and have mostly just repeated other peoples analogies when talking about it, so I can't judge if this analogy is correct.
The explanation:

If you're in a car traveling at a constant speed and the car enters a turn, you are pressed against the side of the car. If you're in a rollercoaster and the track curves upwards you experience being pressed into your chair. (This is a familiar experience for most people and they accept it)
You are currently sitting in a chair, at rest. But now imagine that you are not actually at rest, but rushing into the future with one second per second. You are not at rest. You are moving in time.
It is space-time that is curved according to the model in general relativity. The mass of the earth causes the rollercoaster track you are traveling forward in time on to be curved "upwards". Thus, when you sit at rest in your chain rushing into the future you experience being pressed into your chair.

This makes intuitive sense to me and the non-physics people I have tried the explanation on, but is it correct?
 A: There is a stackexchange question titled: 'How much doe the curvature of space change the volume of Earth by?'
As I understand it, the amount of spatial curvature is such that if it would be possible to measure both the Earth circumference and the Earth radius to sub-milimeter accuracy a small deficit would be measured; the deficit comes out in the region of 1 milimeter of the radius.
On one hand I concur that one can posit there is no such thing as 'being at rest', since one is always traveling forward in time at a rate of one second per second. But: I don't see how to connect that with the spatial component of space-time curvature.


Have you considered the River model of GR spacetime ?
Andrew J. S. Hamilton and Jason P. Lisle (JILA, U. Colorado), present that there is a self-consistent model (up to being mathematically consistent) in which spacetime flows down gravity gradients. In the model objects in spacetime are not affected by the velocity of spacetime. As spacetime accelerates down a gravity gradient object follow the acceleration of the spacetime that thay are in.
Uniform circular motion
When an object is accelerating with respect to spacetime a force is required to sustain that acceleration.
Orbital motion
For an orbit around a celestial body: the trajectory of the orbit curves away from a straight line because the orbiting object responds to the acceleration of the spacetime that it is moving through. Hence being in orbit around a celestial body is locally a state of micro-gravity.
On the surface of a celestial body
When you are sitting on the surface of a celestial body the spacetime you are in is accelerating towards the center of gravity. If the surface of the celestial body would not exert a upward force you would follow that acceleration, towards the center of gravity. The force exerted by the surface on you sustains your height above the center of gravity.
Hamilton and Lisle emphasize that it isn't necessary to take the river model literally. They present the model as a useful tool because of the mathematical consistency. It allows you to think about very different circumstances in a unified way.
Quoting Hamilton and Lisle:

The picture of space falling like a river into a blackhole may seem
discomfortingly concrete, but the aetherial overtones are no more
substantial than in the familiar cosmological picture of space
expanding

A: When a layman thinks about gravity, its normally about how strong or weak the gravitational field is. But curvature of space-time is related to non-uniformity of the field, and not to its intensity.
For instance: at the surface of Jupiter a person would weight 2.5 more than at earth, but the space-time curvature is smaller there than here. 
A better example for such a curvature is an artificial satellite. It is always falling, but never hits the ground. If gravity were uniform, it would follow a parabola, as for a projectile from a cannon. 
Not only absence of gravity, but an hypothetical infinitely flat planet also means flat space-time.  
