They can tell that the big bang is the light source because of Hubble's Law[0]. In short, the universe itself (so space itself) expands[1], and it does so fast enough to create a Doppler effect in light, causing redshift[2].
When we look further out into space, we effectively look "back" into time because of the limit of the speed of light. The older the light, the more it has redshifted because the more space has expanded while it was travelling. So this redshift increases very predictably the further we look out into space and back into time.
As a result, when observing space we can say when light is left-over from the big bang, because we can tell the age of the light by how much its spectral lines have redshifted.
When we look further out into space, we effectively look "back" into time because of the limit of the speed of light. The older the light, the more it has redshifted because the more space has expanded while it was travelling. So this redshift increases very predictably the further we look out into space and back into time.
As a result, when observing space we can say when light is left-over from the big bang, because we can tell the age of the light by how much its spectral lines have redshifted.
[0] https://en.wikipedia.org/wiki/Hubble%27s_law
[1] https://www.youtube.com/watch?v=_llA2q1rlSg
[2] https://en.wikipedia.org/wiki/Redshift