Galaxies are huge collections of stars. When the light from a galaxy is split into a spectrum, it contains dark lines at particular wavelengths, which can be compared with the same lines measured in a laboratory. In light from almost every galaxy, these lines are moved towards longer wavelengths, which is the red end of the spectrum. This is called red-shift. It happens because the galaxy is moving away from us, and that stretches the light waves.
Red-shift is bigger for galaxies that are further away. A bigger red-shift means a bigger change in wavelength, which means the galaxy is moving away faster. So the further away a galaxy is, the faster it is moving away from us.
This pattern is exactly what we would expect if the Universe is expanding. Imagine dots drawn on a balloon that is being blown up. Every dot moves away from every other dot, and dots that start further apart separate faster. The galaxies are like the dots and the stretching space between them is like the rubber. The same pattern would be seen from any galaxy, so it does not put the Earth at the centre of the Universe.
