Electromagnetic radiation from stars in distant galaxies shows
a "red-shift", that is, a shift that makes the wavelengths appear
to be longer than they really should be.
The real trick to this kind of observation is figuring out what the
wavelengths you observe really should be, since you only see them
in their shifted condition.
The only process we know of that can make wavelengths longer
is when the source of the waves is moving away from the observer.
So we assume that this is the cause of all the radiation from distant
galaxies having its wavelengths stretched, and this leads us to the
concept that the universe is expanding. There's no way to prove it,
so the expanding universe is "only a theory" ... just like gravity and
elliptical orbits and nuclear fusion, which also can't be proven.
