How do we know that galaxies are moving away from the center of the universe?

The red shift in the spectrum indicates that the galaxies are red shifted, which is an indication that they are moving apart. The universe has no center.

We would have a point of reference to gauge the speed of galaxies if the universe had a center.

We can only measure the relative speeds of a galaxy with respect to another galaxy, as the theory of relativity effectively asserts the absence of any point of reference.

Although galaxies are moving in all directions, we cannot state that all galaxies are traveling in the opposite direction.

We live in a region we refer to as the visible universe, which contains all of the objects that light has had time to reach us. We will never know what lies outside of this region. The concepts of center and edge have no bearing on the nature of the universe.

Hubble's Law states that the redshift of galaxies in their spectra, which shows that they are moving away, is evidence for the universe's expansion.