If a star is moving away from the earth, we see its light red-shifted, that is the wavelength of the light is lengthened, so that a white star would appear more red. When we look at galaxies we see their light is red-shifted, so we know they are moving away from us. Light from more distant galaxies is more strongly red-shifted, so they are moving away from us more quickly.
We don't believe the earth is special, so people in other galaxies would also see other galaxies moving away from them in the same way we do.
If we imagine time running backwards we would see all the galaxies coming together. Furthermore, galaxies twice as far away are moving twice as fast, so all the galaxies would come together in the same place at the same time. This is consistent with the Big Bang theory, that the universe began with all matter and energy in one place, and expanded from there.
There is much more to the Big Bang and expansion than that, for example the galaxies are not so much moving into empty space as being separated by space itself expanding. However the idea above was fundamental in the origin of the Big Bang theory.