The Big Bang was first proposed by Georges Lemaitre in 1927. Lemaitre was a professor of Physics at Leuven Catholic University in Belgium and a Catholic priest.
His Big Bang theory was based on observational results of Edwin Hubble and theoretical equations developed by Alexander Friedmann, a Russian mathematician who used Einstein's Theory of Relativity to develop mathematical models for an expanding universe (instead of a static one proposed by Einstein).
The theory ran into trouble because its main opponent (Fred Hoyle) was: the best astrophysicist of his days. A very convinced atheist, he was against the theory. In fact, it was not the original name of the theory. It was meant to make fun of the theory. Fred Hoyle could not accept a theory that would suggest that the Universe could have been created, especially since the hypothesis was developed by a priest. Fred is the one who came up with the name Big Bang during a radio interview. One objective was to make fun of the theory. The other was to cause confusion: it worked since it causes a lot of people to think that there was an initial explosion.
The Big Bang was NEVER discovered. It is just a THEORY. No one has ever proven it to be true! In other words, it is just someone's BEST GUESS, as to what happened when the Universe began. But that assumes that the Universe DID begin. Other people think that the Universe always was, and always will be. These people have other theories, which do NOT include a Big Bang.