How does the Theory of Relativity relate to the Big Bang Theory?
Special and General relativity have both contributed the theoretical basis of the the nature of energy, gravity and of spacetime that made the "Big Bang" theory possible.
Special Relativity resulted in the famous
(Here, I am assuming you have a good grasp of the Big Bang and how the universe came to be; I will not detail that lengthy story!)
The part that General Relativity has played is essentially to produce the equations that would be solved to create possible scenarios for how the expansion of the universe took place. Essentially, the equations of General Relativity allow for certain mechanisms to occur when we try to grasp the many events that had to have occurred at various stages in the history of the universe - for instance, the creation of light, the appearance of the first quarks, the first atoms, and so on.
All in all, it is a fascinating topic to dive into! But also much to involved to do more than what I have outlined here. I hope that is what you were looking for!