A few thoughts off the top of my head. 1) the uniformity of the CMB, and the fact that it is a near perfect blackbody cannot be explained through any known galactic process. We would need dramatic new physics to account for it, whereas the big bang model predicts it just as we see it. 2) The CMB encodes the history and contents of the universe in its power spectrum, so it tells us something globally about the observable universe, rather than reflecting local galactic processes. 3) As the CMB travels through the universe, it interacts with matter (ionized gas, hot electrons, etc), and these interactions leave imprints on the CMB. These imprints are found to correlate well with large scale structures, like galaxy clusters, in the universe. That ain't going to happen by chance.