Answer:
The theory predicts an omni-present, isotropic microwave radiation with a spectrum equal to that of a 3.7 K black-body. That is exactly what we detect, and no other hypothesis can explain it other than saying, "It's there but I have no way to explain why."
It predicts that far away galaxies will be much younger than those close to us. This is indeed what we observe.
It predicts a specific ratio of hydrogen to helium to deuterium in every part of the Universe. What we see is exactly as predicted. Again, all other hypotheses can only say, "That's the way it is, but I can't explain why."