The main evidence is the redshift of galaxies. This means the other galaxies are moving away from us. The Andromeda galaxy is having a blueshift though, meaning it's coming closer. Another piece of evidence is warmth in space. Hope this helps.
It has been observed that the galaxies are moving away from each other. This expansion is an outcome of the Big Bang theory of Universe formation. The observational proof of this is the Redshift observed in the spectrum of faraway galaxies which means that they are moving away. Also, as per Hubble Law, the galaxies which are farther away from us are moving faster as compared to the ones which are closer.
