Some technologies used to find evidence supporting the Big Bang theory are telescopes and satellites. Edwin Hubble himself used a powerful and new telescope (for 1929) that helped him understand the universe was bigger than previously thought. There are large amounts of satellite data that back up the theory, such as the data from the Hubble Space Telescope. The COBE satellite led to observations leading to the discovery of dark matter, which is the matter that seemed unexplainable (and was a hole in the Big Bang theory).
Picture tooken by the Hubble Space Telescope.