In 1929, Edwin Hubble discovered that most of the universe was expanding and moving away from everything else. If everything is moving away from everything else, then in past times things were closer together. For several decades the data has seemed to say that the recession velocity varies linearly with distance. Extrapolating that backwards in time suggested a time when the whole universe was very small and tightly packed.
About 300,000 years after the Big Bang, enough energy condensed into matter to make the universe fairly transparent, and high speed photons were racing in all directions. They may still exist in the form of the Cosmic Microwave Background radiation. It's an independent measurement that corroborates Hubble expansion.
Another corroboration is the percentage of H2 and He in the universe; they are consistent with the form of expansion the big bang predicts.
OK, there's a wrinkle called inflation in the first tiny fractions of a second; see diagram below.
Recent measurements seem to show an increase in the expansion rate. These go against the standard inflation plus big bang theory. This acceleration also goes by the name "dark energy." and is being enthusiastically investigated.
