This is evidenced by Doppler shift effect. This is the phenomenon where waves that are coming from an object that is accelerating towards your direction are heard at a higher frequency than that of objects moving away from you. This is the same for light (which also has wave-like properties). When observing distant galaxies, astronomers discovered that they all showed to be redshift (change in the frequency of light towards red color). This means that all galaxies are moving away from each other because the universe is expanding.