The evidence that supports the idea that Cosmic Microwave Background (CMB) radiation is a remnant of the Big Bang is:
Its temperature is uniform.
The uniform temperature of the CMB across the universe suggests that it is a remnant of a hot, dense state that existed shortly after the Big Bang, when the universe was much smaller and more homogeneous. Later, small fluctuations in temperature give insight into the density variations that led to the formation of large-scale structures in the universe.