The Big Bang model says that the universe began about 14 billion years ago from a single point that was extremely small, hot and dense. Space itself has been expanding ever since, and the universe has cooled as it has grown.
Red-shift supports this model. Distant galaxies are moving away from us, and the further away they are, the faster they move. If the expansion is run backwards in time, all the matter in the universe comes back together at one point. The speeds fit the idea that everything started from the same place at the same time.
The second piece of evidence is cosmic microwave background radiation (CMBR). This is low energy microwave radiation that reaches Earth from every direction in space. The Big Bang model explains it as radiation released in the early universe, when everything was very hot. As space expanded, the wavelength of that radiation was stretched, and today it lies in the microwave region. Earlier ideas of a universe that never changes cannot explain the CMBR, so the Big Bang model is the one accepted by most scientists.