The first thing to think about is what exactly we mean by a big bang. A weak version of the big bang hypothesis would simply be the statement that at some time in the past, the universe was extremely hot and dense -- as hot and dense as a nuclear explosion. A stronger statement would be that, at some point in the past, there was a singularity, which is in nontechnical terms a beginning to time itself.
We have a variety of evidence that the universe’s existence does
not stretch for an unlimited time into the past. One example is that in the present-day universe, stars use up deuterium nuclei, but
there are no known processes that could replenish their supply. We
therefore expect that the abundance of deuterium in the universe
should decrease over time. If the universe had existed for an infinite
time, we would expect that all its deuterium would have been lost,
and yet we observe that deuterium does exist in stars and in the
We also observe that the universe is expanding. There are singularity theorems such as the Hawking singularity theorem and the Borde-Guth-Vilenkin singularity theorem ( http://arxiv.org/abs/gr-qc/0110012 ) that tell us that, given present conditions, there must be a singularity in the past. These theorems depend on general relativity (GR), which at this point is a well tested, fundamental theory of physics with little viable competition. Although there are competing theories, such as scalar-tensor theories, observations constrain them to make very nearly the same predictions as GR under a broad range of conditions.
There is a little bit of wiggle room here because the Hawking singularity theorem requires a type of assumption called an energy condition (specifically, the strong energy condition or the null energy condition), and BGV is more of a model-dependent argument having to do with inflationary spacetimes (which violate an energy condition during the inflationary epoch). An energy condition is basically a description of the behavior of matter, sort of roughly saying that it has positive mass and exerts positive pressure.
Dark energy violates the standard energy conditions. So if dark energy is strong enough, you can evade the existence of a big bang singularity. You can get a "big bounce" instead. However, we have three different methods of measuring dark energy (supernovae, CMB, and BAO), and these constrain it to be too weak, by about a factor of two, to produce a big bounce. The figure below shows the cosmological parameters of our universe, after Perlmutter, 1998, arxiv.org/abs/astro-ph/9812133, and Kowalski, 2008, arxiv.org/abs/0804.4142. The three shaded regions represent the 95% confidence regions for the three types of observations. If you take the intersection of the three shaded regions, I think it's pretty clear that we're just nowhere near the region of parameter space that results in a big bounce.
There are various other observations that verify predictions of the big bang model. For example, abundances of light elements are roughly in agreement with calculations of big-bang nucleosynthesis (although there are some discrepancies that are not understood). The CMB is observed to be very nearly a perfect blackbody spectrum, which is what is predicted by big bang models. This is hard to explain in models that don't include a big bang.
Historically, cosmological expansion was observed, and cosmological models were constructed that fit the expansion. There was competition between the big bang model and steady-state models. The steady-state model began to succumb to contrary evidence
when Ryle and coworkers counted radio sources and found that they
did not show the statistical behavior predicted by the model. The CMB was the coup the grace, and the big bang model won. When you observe the CMB, you're basically looking up in the sky and directly seeing the big bang.
Note that although in historical cosmological models, perfect symmetry was originally assumed in the form of homogeneity and isotropy, to make models easy to calculate with, these are not necessary assumptions. The singularity theorems do not assume any special symmetry. For the Hawking singularity theorem, you just need to have a positive lower bound on the local value of the Hubble constant, and that bound has to hold everywhere in the universe on some spacelike surface. Of course, we can't observe all of the universe, and you could say that our reason for believing in such a global bound is homogeneity. However, the existence of such a bound would be only a very, very weak kind of homogeneity assumption -- much weaker than the kind of symmetry assumptions made in specific models such as ΛCDM.