What evidence do we have for the change in size of the universe?

Edwin Hubble discovered the first indication of this when he noticed that certain lines in distant galaxies were redshifted, indicating that they were traveling away from us.

A galaxy moving towards us will have more blue lightwaves as the wavelength is lowered by the galaxy's movement; the light from a galaxy moving away from us will appear redder as its waves are stretched out (wavelength increased), which is called redshift. Redshift is a result of the Doppler effect. For example, when an ambulance is speeding towards you, its pitch appears higher as the sound waves are compressed. As it moves away, the pitch lowers as the sound waves are stretched out.

Redshift caused the light that Hubble actually saw to be redder than what he should have seen, indicating that the galaxies were moving away from us.

The universe was expanding because the redshift was higher for galaxies that were farther away from us. This has been compared to an expanding balloon: when a balloon is blown up, all of its points move away from the center, but a point on its surface will move away from the center more quickly. Hence, just as the balloon is expanding, so too are the outer reaches of the universe expanding!

The theory of cosmic expansion, which suggests that the universe is expanding over time rather than remaining static, is supported by evidence for the change in universe size, including measurements of the redshift of distant galaxies, observations of the cosmic microwave background radiation, and the apparent acceleration of the universe's expansion observed through Type Ia supernovae.