How do we know that galaxies farther away from us are moving faster than nearby galaxies?
Astronomers can observe light from distant galaxies. When they do this, they see it is different to the light from the Sun. The dark lines in the spectra from distant galaxies show an increase in wavelength. The lines are moved or shifted towards the red end of the spectrum. This effect is called red-shift. The diagram shows part of the emission spectrum of light from a distant galaxy. Red-shift and speed Astronomers see red-shift in virtually all galaxies. It is a result of the space between the Earth and the galaxies expanding. This expansion stretches out the light waves during their journey to us, shifting them towards the red end of the spectrum. The more red-shifted the light from a galaxy is, the faster the galaxy is moving away from Earth. Source/ further reading: https://www.bbc.com/bitesize/guides/zstb8mn/revision/1
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