How does cosmological redshift relate to the expansion of the universe?
It shows that everything in the universe is moving away from each other; the Universe is growing.
Edwin Hubble was not the first person to suggest a non-static model of the universe. However, he was the one to publish measurements that gave solid ground to this theory. Hubble noticed that when he observed other galaxies they were redshifted.
Redshift occurs because red light has the lowest frequency of all the colors, and when something is moving away from you its spectral lines are all moved more toward the red. (Alternately, if something is moving toward you, it becomes blueshifted with a higher frequency).
Think of a motorcycle riding down the street your're standing by. As the motorcycle comes closer and closer the sound becomes louder and louder due to an increased frequency in pitch. Then as it passes you and continues racing down the road, you notice the sound becoming lower in pitch; the frequency is lowering with increased distance.
Remember, light behaves like a wave, as does sound. So just like that motorcycle in the above analogy, when galaxies recede from us they will display a lower frequency in their color, leading to a red tinge or "redshift". When Hubble looked out and saw ALL of the galaxies receding from us and from each other, it proved that the universe had once been much smaller, with objects closer together. Therefore, the universe is expanding.
Sorry if the answer was a little long. I just wanted to make sure you had the full understanding. Good luck!