The red shift is a consequence of the Doppler’s Effect, a phenomenon wherein waveforms that move away from an observer observe an increased wavelength - and hence decreased frequency - due to its relative motion:
Now, how do we apply this logic to the Big Bang Theory? When it comes to space, we’re essentially capturing the visible light waves (and possibly other electromagnetic waves) coming from faraway star systems. Upon receiving them, we realise that these observed wavelengths have shifted towards the “red side” - i.e. the region of the electromagnetic spectrum with longer wavelengths. This implies that these stars are moving away from us naturally. This corroborates the Big Bang Theory, which states that the universe actually expanded from an initial point of high density and temperature, and then has been expanding away at a fast rate ever since.