If we extrapolate far enough back, we reach a point where the predicted density would be infinite. We call that point the beginning of time (or the big bang). This "prediction" of an infinite density is taken as a sign that our model breaks down at (or before) this point (by "before" I mean when we extrapolate backwards in time towards the infinite density time...).
So, it works extremely well for a time when the density was extremely large (but not infinite) until today. We just don't have enough information to know what would happen when densities larger than a certain value so as to make meaningful predictions.
To make a completely silly analogy: it would be like predicting the motion of a rocket aimed for a comet. Our knowledge of celestial mechanics is good enough to predict the trajectory up until the rocket "touches" the comet. We just do not have enough information about the comet composition to know if the rocket will just blow it apart or if it is the rocket itself that will splatter on the comet, or whatever.
The best experimentally-verified cosmological models we have point to the fact that the Universe was in a hot and dense state a few microseconds after a specific point of time about 13.7 billion years ago. Extrapolating those few microseconds back in time we get an unphysical result from our current models, but that simply means those models are inaccurate if applied to the beginning of time.
Contrary to the popular belief, The Big Bang model simply does not concern itself with what happened at the exact moment of Big Bang; what it does is describe the evolution of the Universe after that, and does it well.
That sounds as much of a cop out as the answers religious people give.
That's wrong. As soon as the universe's mass/energy expanded sufficiently for there to be a time dimension (IOW as soon as the universe was something other than a singularity), the present physical laws applied. Some of the physical constants with which we are familiar may have had different values, but the same set of rules have been in effect since there were four distinct dimensions. This is borne out by the existence and properties of the CMB, as well as other observations.
Your use of the term "big bang time" implies that there was, in fact, time, which means four dimensions existed, which means our present physical rules were in effect.
One idea has it that the natural forces we evaluate separately now, were unified by the conditions of the early universe, but this idea doesn't contradict the notion that basic physical principles were in any way different then.
One of the more interesting cosmological ideas is that the positive mass/energy of the universe is exactly balanced by negative gravitational potential energy, as the universe expands and continuing to the present, but only at one specific expansion velocity: what we call escape velocity. If this idea is true, it means the universe could have sprung into existence spontaneously, like a virtual particle pair, without violating the law of conservation of energy. Stephen Hawking explains this idea in his book "The Grand Design": "Because there is a law such as gravity, the universe can and will create itself from nothing. Spontaneous creation is the reason there is something rather than nothing, why the universe exists, why we exist. It is not necessary to invoke God to light the blue touch paper and set the universe going."
The Big Bang is widely adopted in light of this. Preceding the expansion there has to be a state of greater compression, but there is no explanation on how this state could be possible.
The fact that there is an ongoing process of scientific discovery that has continued unabated, while theological knowledge remains stagnant or even decays, is alone enough reason to give credence to even the most far-fetched theory that might one day have a hope of being tested.
To summarize, hard science like theoretical physics is not about beliefs or answers or end goals, it's about a continual process of expanding knowledge, guided outward by the imaginations of theoreticians.
As I understand the process, theoreticians will invent mathematical constructs that explain everything we have measured already, plus make predictions about something else we can test. If all of the testable predictions are verified, then we are somewhat safer in accepting any untestable predictions of the same model. Then, an expanded mathematical theory will be devised with new physical tests, etc. ad infinitum.