As far as we can tell, the expansion of the Universe started many billions of years ago from a very hot, very small state. From that hot, small state, it mushroomed and evolved into the Universe we know today. Cosmologists call that process of expansion the Big Bang because at some phases, especially in the beginning, the process was rather like an explosion.
Much of understanding the Big Bang is extrapolating between knowledge of particle physics today, and projections from the mathematical model of an expanding universe in general relativity. The Einstein equations give us a mathematical model for describing how fast the Universe would expanding at what size and time, given the energy density of matter and radiation at that time. We base our guesses about the matter and radiation density of the early Universe based on the ancient light reaching us from the past in our night skies, and what we have learned about elementary particle physics, through theory and experiment.

The Big Bang Tour starts here