Two pieces of evidence support the Big Bang theory: the red-shift of light from galaxies and the cosmic microwave background (CMB) radiation.
Light from distant galaxies has a longer wavelength than we expect, so it is shifted towards the red end of the spectrum. This is called red-shift. It shows that galaxies are moving away from us, and the further away a galaxy is, the greater its red-shift. This suggests the Universe is expanding, which fits with it having started from one small point.
The CMB radiation is microwave radiation that reaches the Earth from all directions in space. It is almost the same in every direction. It is thought to be radiation left over from the very hot early Universe. As the Universe expanded, the wavelength of this radiation was stretched until it became microwaves.