Explain how red-shift provides evidence for the Big Bang theory.

Red-shift describes the phenomenon whereby the wavelength of light from a source increases as it travels through spacetime, hence observed as being 'shifted' to the red end of the spectrum (as red light has the longest wavelength in the visible spectrum). The red-shift of light from galaxies we observe can be used to determine these galaxies' recessional velocities (the speed at which they are moving away from us), as well as their distance.
Using these red-shift observations, we can determine that galaxies are moving away from us with increasing recessional velocity, indicating that the universe is expanding. If this is the case, there must have been a point in time at which the universe began expanding, when all matter and energy was converged upon a single point (called a singularity): this is the fundamental principle on which the Big Bang theory is based.