We need one step back here.

Physics use mathematics to describe and create theories. So let’s move for a moment to the mathematics world.

In the mathematics world, curved space is a well defined notion. A quick search with google and I got this: https://en.wikipedia.org/wiki/Curved_space

Curved space often refers to a spatial geometry which is not “flat” where a flat space is described by Euclidean geometry. Curved spaces can generally be described by Riemannian geometry…

A circle or the surface of a sphere is an example of a curved space. They don’t need an extra dimension to curve in. Have a look to the link above, section “Simple two-dimensional example”.

I used the plain example because you mentioned it. An hyperplane is just the generalization of a plane. Remember, we are still in the mathematics world.

Now let’s move toward the physics world. From the same article above:

Curved spaces play an essential role in general relativity, where gravity is often visualized as curved space.

According to the general relativity theory (note the word theory), masses curve the space-time around them. It is important to understand that it is not just space, but space-time. What it means is that the mathematical concept of curved space-time (have a look to https://en.wikipedia.org/wiki/Minkowski_space) seems to apply very well to what we observe. Up to now general relativity has not been falsified, even though we know that it doesn’t work at micro scales, where quantum theories explains some of the strange effects that happen at micro scale level. But it works very well at cosmic scales. Black holes are an example of a general relativity prediction, but they may be an example where the Einstein’s equations fails because of the quantistic effects.

In short, curved space-time it is a mathematical concepts which works well in describing the effects of masses in the space that we measure.

Up to now we have a space-time, not a space-time fabric. So why is Ethan using the word “space fabric”? Because of some quantum theories and because we have some evidence that suggest that the space itself it is expanding. Again it is a mathematical concept that well explains the fact that the most distance is a galaxy from us, and the most fast it seems to move relatively to us. Some galaxies moves apparently faster than light relatively to our point of view.

I mentioned quantum theories defining space fabric. It seems that there is an intrinsic energy related to space itself. I found some interesting answers to the question “Does space itself have an inherent energy?” here: https://www.quora.com/Does-space-itself-have-an-inherent-energy

So, if I understand properly what Ethan says, he uses the words “space fabric” to refer to a curve dspace-time, expanding, full of energy left over from the big-bang and the inherent energy that predicts the

So-called “virtual particles” (original meaning: short lived quantum entities)…

Finally, have a look to this article about clock synchronization, especially the conclusions: http://galileoandeinstein.physics.virginia.edu/lectures/synchronizing.html