Why does red shift support the big bang theory?

Because if everything we can see is moving away from us, and the further away the faster it's going, if you extrapolate back in time you realize that at some point everything must have been very close together. That's more or less the core of the Big Bang theory.

answer2:

The Big bang is a bust if there is no expansion , motion! Red shift is interpreted as the motion, thus the Big Bang is possible. However, the red shift is misinterpreted. Hubble himself, did not accept the red shift as recession, rather Hubble called the red shift, "a hitherto unrecognized principle of Nature".

The red shift is the indicator of the centrifugal force, associated with the "Dark Energy", the vector energy Ev=mcV. Properly interpreted, the red shift indicates no Expansion, no Big Bang.