How do we know what happened during the Big Bang?

This seems to be a common misconception about the big bang.

At present our theories can only suggest what happened AFTER the "bang". We cannot formulate what occurred AT the singularity with our current knowledge of physics.

At a small neighborhood around a spacetime singularity quantum gravity becomes important and we simply have no clue at present how to deal with the general relativistic singularities (e.g. black holes as well!)

When you hear of physicists talking about the big bang it is almost always a discussion of the dynamics AFTER the singularity.

When physicists discuss what happened during the planck era or at a singularity it is entirely (very) educated guesswork within the bounds of current theory (much like how theoretical physicists like hawking have come up with convincing theories about some black hole properties).

Hope that helps!

We know some physics that occurred with the early universe up to periods of time close to the big bang. The actual moment of the big bang is not known at least empirically. That moment did not occur at point, but was a process where the spacetime of the observable universe emerged. This was probably a bubble nucleation event similar to that proposed by Coleman and de Luccia. A quantum field at a high energy vacuum tunnels through a barrier and emerges as a "bubble of spacetime." This and related variants are thought to be involved with the earliest moments in the observable cosmology.

This tunneling process converts a spacetime region, which is an de Sitter spacetime or anti-de Sitter spacetime, from one with a very large cosmological constant to one with a very small one. This is similar to radioactive decay and the energy gap between the two vacua generates particles and radiation. The de Sitter spacetime is rapidly inflating due to a large cosmological constant and is associated with the inflationary period. Data from WMAP and Planck spacecrafts have found anisotropic distributions in CMB radiation. The early anisotropy of the universe left imprints on the much later period where the radiation dominated phase ended $380,000$ years into the evolution of the universe

There are other more recent forms of inflationary cosmology with chaotic inflation, where the bubble emerges from an "eternally inflating" de Sitter spacetime and is but one of an infinite number of cosmologies. It is curiously similar in a way to the old Hoyle-Bondi theory of continuous creation in a steady state cosmology. What we observe as a big bang is just one of an infinite number of "creation events."

After this time period predictions with the number of quarks, where more than three families would increase the effective heat capacity of the early quark plasma, conform to just three families. Other predictions such as deuteron and helium production in the first three minutes of the universe are also supported by data.

There are then ways in which we can observe moments prior to the CMB event or the end of the radiation dominated period. The radiation in the CMB is itself thought to be a detector of its own, such as holding imprints of gravity waves produced during inflation. B-modes are finger prints of gravity waves, and in 2014 the BICEP-2 announced their discovery. However, subsequent concern with EM polarization from galactic dust has reduced the experimental certainty below 5-sigma, so the search is in effect still on.