Relationship of red shift and the big bang theory?

The Big Bang theory predicts an expanding universe. The red shift, being the Doppler effect of light when something is receding from you, confirms this because when looking at galaxies the light emitted from them is red shifted. The farther away galaxies have a higher red shift, confirming that all galaxies are moving away from each other, not just the Milky Way (i.e. the universe is expanding).