What does the curved nature of space-time,...?

The universe is expanding (space itself is expanding, not just "the distance between things").

The further out we look, the further back in time we see.

For example, when I look at a galaxy that is 3 million light-years away (a very close neighbor), I see it as it was three million years ago, because that is how long the light took to get here.

In a "flat world", If I look at a point A which is 10 miles one way, and a point B which is 10 miles the other way, I would expect the distance from A to B to be 10+10 = 20 miles.

For short distances in the universe, the same rule seems to work (at least, close enough).

However, if I look at a point "A" on the Cosmological Microwave Background radiation, 13.7 billion light-years away in one direction, then I observe a point B on the CMB in the exact opposite direction, I see each point as they were 13.7 billion years ago.

At that time, A and B were separated by a much smaller distance. Something like (at most) 30 million light years.

30 million is a lot smaller than 13.7 + 13.7 = 27.4 billion light-years. How is that possible?

That is because I see A and B as they were, not as they are. The space-time representation of the universe appears curved to us, because we do not see all its parts "at the same moment in time". The further out we look, the smaller the universe was at the moment we see it.

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Here is an analogy. As usual, remember that analogies are not perfect comparison. They serve only to show a principle, and the principle here is "curvature", not time, nor space-time.

If Earth's surface were flat, we'd expect the distance from A to B to be simply the sum of the distances from me to A, and from me to B.

Stand at the North pole (OK, just imagine you are).

Pick a direction, and imagine that you see an object "A" located at 10,000 km (6,250 US miles). That object is at the equator. Turn completely around (180 degrees) and imagine object "B", also at 10,000 km.

You expect that the distance between these two objects should be 10,000 + 10,000 = 20,000 km.

If a person at object A were to go out and measure the distance separating them from B, they would find... 20,000 km.

So far, no surprise.

Now, move both objects away from you; that means that they are both moving south.

Move them by 5,000 km each.

This means that the distance from you to A is 15,000 km, and the same distance from you to B.

Total = 30,000 km.

Yet, if a person on A goes out towards B, they find the distance between them is now only 10,000 km. It has gone down!

Push them out another 4,000 km. Each one is now 19,000 km from you, in opposite directions.
The real distance between them is now only 2,000 km.

Sure enough, if you keep asking them to move away from you, the distance between them will get to zero (they will collide at the south pole). At that moment, A is 20,000 km from you in one direction, and B is 20,000 km from you in the other direction.

You are trying to tell them that they are 40,000 km apart, while they are busy determining how they are going to explain this to the insurance company...

Space-time is by itself flat. However, matter is present and that puts a tension on the fabric of space-time creating a curvature depending on the amount of mass. Since matter is also natural, we can say the nature of space-time is curved.

It means if you started out from the earth in what you perceived to be a straight line in one direction and kept going, eventually you would return to your starting point.

Which means, even if you draw the straightest line possible on a piece of paper, there's still an imperceptible "curve" to it due to the curved aspect of "space and time..."

Well, ..., the Universe as a whole, according to recent data from the WMAP probe, is not curved. It is flattened (See the links below). Now this data is not without controversy. Do you mean to tell be that after 518 (See Columbus - 1492) years of exploration and discovery mankind is still thinking with a flat mentality? Well, All I'm saying is that current data and the most recent experiments point to such an understanding. One thing for certain is that space is not Spherical or Hyperbolic. However, it could be an oblate spheroid and that would account for and support a yet undetected curvature of space without contradicting current data. Only time and more study will tell.

Finally, that does not mean that space immediately around us can not be warped or curved. In fact it can due to massive gravitation. This fact was conclusively demonstrated during the solar eclipse of 29-May-1919. See the last link below.

John Wheeler put it succinctly when he said, "Matter tells space how to curve, and space tells matter how to move." (He could have said "space-time" rather than "space," but to me that would sound less poetic.) A light ray is the shortest distance between two points. Near a massive object, a light ray will be bent, but if you try to straighten it, it will get longer. That's what is meant by the curved nature of space-time. Although, as has been pointed out, this works only on a local scale. On a cosmic scale, space-time appears to be flat.

Gravity warps the space-time fabric where an object resides.

The scene of curved coordinates is for visualisation purpose.