How does General Relativity relate to the Big Bang?
The Standard Model of Cosmology (popularly known as the Big Bang Model) is built on the fundamental assumption that gravitation as modelled by Einstein's General Relativity is a correct.
As gravitation is the most dominant force on cosmic scales, construction of cosmological models require a model for gravitation. The Standard Model of Cosmology (popularly known as the Big Bang model) is based on General Relativity, which is Einstein's model of gravitation.
Newtonian Model of Gravitation would not allow the construction of cosmological model because of its action-at-a-distance feature. Unless the mass in the Universe is so uniformly distributed and the Universe is infinite in extent in all directions, all masses will collapse to a point according to Newtonian Model of Gravitation. So Cosmological models were impossible in Newtonian Gravitation.
General Relativity modelled Gravitation in terms of the geometry of space-time. This allowed the possibility of a dynamic Universe which could be expanding (or contracting) preventing the gravitational collapse of all matter to one point. General Relativity forms the foundation on which the Standard Model of Cosmology is built.