With the Doppler effect you can determine the speed along the line of sight of something that emits light or another kind of electromagnetic radiation. If you know how much light something emits and you can measure how bright it appears from here, then you can calculate how far away that thing must be to look that bright (or dim) with that amount of produced light. By combining those methods, Edwin Hubble could make a graph in 1929 that showed how the Doppler speed of nearby galaxies depended on their distance. The measurements turned out to lie near a straight line, which meant that the further away a galaxy is from us, the faster it moves away from us. This relationship is now called Hubble's Law. Nowadays we can measure distances and speeds of galaxies that are much farther away than the farthest ones Hubble could investigate, and the Law holds also for those very far galaxies.
Different explanations are possible for these observations. One explanation is that all galaxies (or at least the matter from which they formed) was clumped together a long time ago at one place in otherwise empty space, and that an explosion happened which hurled material into all directions at different speeds. The fastest material has traveled furthest, but the slowest material did not get far, so there is a relationship between the speed of the material and its distance from the point of origin. In such a Universe there is a special place, namely the place where the explosion happened. Only observers in such a place see a Hubble's Law, as we do. In such a Universe we'd have to be in just that very place, or else our observations wouldn't fit.
When the true nature of the Sun and planets and things outside of our Solar System were yet unknown, people took it as self-evident that the Earth was in the center of the Universe (in the Geocentric World View). Each new astronomical discovery that made the location of the Earth less important was denied and obstructed, but to no avail. We now know that the Earth is a small planet orbiting an ordinary star somewhere in the outer regions of a normal galaxy at the edge of an unremarkable supercluster. This makes it hard to believe that at the largest scale we would turn out to be in the special center of the Universe after all.
Another argument against such a privileged position is that our cosmic surroundings do not look clearly different from the rest of the Universe. As far as we can tell, the Universe looks just about the same everywhere, with about the same average mass density and the same distribution of matter over the same types of galaxies. If there had been a giant explosion in our location then you might expect to find some traces of that which you couldn't find elsewhere, but there are no such traces.
There is another explanation for Hubble's Law, namely that the Universe itself is expanding. A long time ago all matter was pressed together, not because it was in one place in otherwise empty space, but because the Universe itself was very small then. Because the Universe expands equally fast everywhere, more space appeared between all galaxies so they ended up further apart, just like small spots on a balloon that is being inflated. The nice thing about an expanding Universe is that you automatically get the same Hubble's Law for everybody. From any galaxy in an expanding Universe it looks as if all other galaxies move away from it at a speed that is larger for galaxies that are further away. You can check this for yourself with the balloon with spots that I mentioned before. Pick a spot at random on the balloon and then inflate the balloon. All other spots seem to move away from the chosen one, but the same holds also for any other spot that you pick the next time.
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