How does a redshift give evidence to the Big Bang Theory?
The red shift indicates that the universe is expanding outward at an ever increasing rate of expansion.
When Hubble determined that the vast majority of stars and galaxies had a red shift, he created a major shift in the world view or paradigm that dominated science. The universe before Hubble was thought to have eternally existed, a so called steady state universe.
After Hubble the presently observed universe could no longer be consider to be eternal.
A red shift in light indicates that the object is moving away from the observer. This is called the doppler shift. or effect. Stand by a railroad track and listen to the train whistle. As the train is moving toward you the sound is higher ( the waves are closer together) when the train passes the sound changes to a lower pitch ( the waves are further apart) blue light has waves that are closer together and red light has waves that are further apart.
The logic is that if the universe is observed to be moving apart it must have been closer together in the past. The big bang theory is that the universe is moving apart from a central point that once existed. In classical Big Bang theory this central point is proposed to have been a superdense ball of matter. The superdense ball was so dense that it was unstable and exploded. This explosion caused the universe that we presently observe.