It was Stephen Hawking who realised that this theory was correct. He had a light-bulb moment when he considered trains going forward and reversing. He associated this with matter in the universe realising that such movement would cause collision - 'big bang' - Essentially, the universe was once concentrated with intense heat and power into a tiny part ('singularity').
It suddenly began to expand. Galaxies started moving away and there has been continuing expansion. The big bang theory can be proved by the fact that the galaxies furthest from us travel fastest, just as happens in an explosion.