I am going to rephrase your question as: How do we know that the universe is moving?
In the 1920's, astronomers discovered the redshift of galaxies. With the development of better telescopes, what astronomers thought were fuzzy stars were actually entire galaxies made of billions of stars. All the galaxies we can see are outside our own galaxy, the Milky Way. Spectroscopic studies of the light from these galaxies showed redshifts. That is, the vast majority of galaxies are moving away from us in all directions.
A redshift is an apparent "stretching" of light waves towards the red, or lower frequency, part of the electromagnetic spectrum. This occurs when light-emitting objects move away from us. Consider, as an example, a police car siren. When the car moves towards us, the pitch (or frequency) of the siren appears to increase. When the car moves away from us, the pitch decreases. This is known as the Doppler effect. Now, the siren did not actually change. It only seemed to change due to its motion relative to us, the observer. Likewise, galaxies that move away from us show redshifts and galaxies that move towards us show blueshifts. This data led to the development of the Big Bang Theory.
If we observe that the universe is expanding, and if we imagine the reverse occurring, then we must conclude that, at some point in the past, the universe began as a single, incredibly tiny point. About 10 to 15 billion years ago (astronomers are still debating about the exact number), the universe suddenly "sprang to life". We don't know why. We don't know what existed before the Big Bang. But, we do have evidence for the Big Bang itself in the leftover microwave radiation that permeates the entire universe. NASA made detailed observations of this background radiation using its Cosmic Background Explorer (COBE).
So, in a nutshell, we know the universe is moving because we observe redshifts in the light coming from distant galaxies.