Hubble discovered the expansion of the universe, but De Sitter, Friedman, & Lemaitre used Einstein's general relativity to predict the expanding universe & the big bang.
Albert Einstein's general theory of relativity predicted that the universe is expanding. Einstein did not believe this implication of his own theory, so he added a term in his equations that he called the cosmological constant.
This cosmological constant was basically a fudge designed and inserted into the theory for the sole purpose of keeping the theory from predicting that the universe is expanding. About a decade later Edwin Hubble observed the red shifts of distant galaxies and discovered that the universe is indeed expanding.
Einstein, to his credit, visited Hubble, examined the data, and admitted to the biggest blunder of his life. Hubble's work convinced Einstein that the universe is expanding after all and that his cosmological constant was not needed to keep his general theory of relativity from predicting that the universe is expanding. Ironically Einstein missed out on getting credit for predicting that the universe should be expanding because he did not believe the implications of his own theory.
There were however a few who even prior to Hubble's work had the faith in the cosmological consequences of general relativity that Einstein lacked.
Willem de Sitter was a Dutch astronomer who in 1917 used Einstein's theory to work out a mathematical description of the universe. In de Sitter's purely theoretical description of the universe, motions in the universe produced the effects of redshifts from large distances. De Sitter did this work before Hubble's redshift work, but it was after Vesto Slipher had noticed the tendency for the spiral galaxies to have redshifts.
De Sitter's description explained these redshifts but it had a fatal flaw. There was no matter in this hypothetical universe. It therefore could not describe the real universe. It was however a good first attempt.
Five years later Alexander Friedman discovered an error in Einstein's solutions and worked out his own correct solutions to the general relativity equations. Friedman described two possibilities for the universe that were consistent with general relativity. One was expanding, the other contracting. Both were realistic enough to contain matter. Friedman also showed that it was impossible using theory alone to tell which description corresponded to the real universe.
Abbe Georges Lemaitre was a Belgian priest and mathematician who is rightly considered the father of the big bang theory. He did not use the term "big bang", but in 1927 he used general relativity to sow the seed that grew into the full big bang theory.
Before Hubble observed the expansion of the universe, Lemaitre described a possible universe that was expanding from its initial state as a single point, a primeval atom about the size of the solar system. Before this expansion started, the universe did not exist. Starting from a singular point, Lemaitre's universe has a specific time at which it was created.
This theory therefore allows, but does not require, the existence of a creator. Does such a creator exist? This question can not be answered scientifically. Being a priest, Lemaitre presumably answered this question theologically rather than scientifically. As there is still no scientific answer, we must each find our own answer to the theological question.
As often happens in science, data such as Slipher's and Hubble's were needed to show that the universe is expanding rather than contracting. Don't ask what the universe is expanding into. As hard as it is to grasp, the universe is all that exists, so there is nothing for it to expand into.
Edwin Hubble correctly gets credit for discovering that the universe is expanding, but Einstein's general relativity theory does predict this expansion theoretically. The big bang theory for the origin of the universe was one response to the knowledge, from both observations and theory, that the universe is expanding.
Barrow, J.H. and Silk, J., The Left Hand of Creation, Oxford, 1983.
Silk, J., The Big Bang, Times Books, 2000.
Harrison, E.R., Cosmology The Science Of the Universe, Cambridge, 1981.
Friedman, H., The Astronomer's Universe, Norton, 1998.