The simple statement "something can not come out of nothing" is, in itself, not very convincing. From quantum field theory, we know that something does indeed come from nothing: to wit, "vacuum fluctuations". In the simplest case, an electron, a positron and a photon can appear effectively out of nowhere, exist for a brief time and then annihilate, leaving no net creation of mass or energy. Experimental support for this sort of effect has been found from a number of different experiments. See, for instance, the Wikipedia page for the Casimir effect.
The common point for all of these effects is that they do not violate any known conservation laws of physics (e.g., the conservation of energy, momentum, and charge). Something can indeed come out of nothing as long as these conservation laws permit this. But people often argue that the Big Bang theory violates the conservation of energy (which is essentially the first law of thermodynamics).
There are several valid counterarguments against this: first, as already pointed out, the BBT is not about the origin of the universe, but rather its development with time. Hence, any statement that the appearance of the universe "out of nothing" is impossible has nothing to do with what the BBT actually addresses. Likewise, while the laws of thermodynamics apply to the universe today, it is not clear that they necessarily apply to the origin of the universe; we simply do not know. Finally, it is not clear that one can sensibly talk about time "before the Big Bang". "Time" is an integral part of our universe (hence the GR term "spacetime") - so it is not clear how exactly one would characterize the energy before and after the Big Bang in a precise enough way to conclude it was not conserved.
Assuming we have some way to handle notions of time outside of our spacetime, the universe appearing out of nothing would only violate the first law of thermodynamics if the energy beforehand were different from the energy afterwards. Probably all people will agree that "nothingness" should have an energy of zero; so the law is only violated if the energy of the universe is non-zero. But there are indeed good arguments that the energy of the universe should be exactly zero!
This conclusion is somewhat counter-intuitive at first sight, since obviously all the mass and radiation we see in the universe has a huge amount of associated energy. However, this tally ignores the gravitational potential energy within the universe. In the Newtonian limit, we can get a feel for this contribution by considering the standard example of a rocket leaving the Earth, with a velocity great enough to "escape" from its gravitational field. Travelling farther and farther away from the earth, the velocity of the rocket becomes smaller and smaller, going to zero "at infinity". Hence the rocket has no energy left "at infinity" (neglecting its "rest energy" here, which is irrelevant for the argument). Applying conservation of energy, it follows that the energy of the rocket was also zero when it left Earth. But it had a high velocity then, i.e., large kinetic energy. It follows that the gravitational potential energy it had on the Earth was negative. For another explanation, see e.g. this post about Negative gravitational energy.
In a Nature article in 1973, E. Tryon sketched an argument that the negative gravitational potential energy of the universe has the same magnitude as the positive energy contained in its contents (matter and radiation), and hence the total energy of the universe is indeed zero (or at least close to zero).
Part of the difficulty here is that the concept of "gravitational energy" is essentially a Newtonian one. In GR, the principle of equivalence makes defining a gravitational energy that will be coherently viewed from all frames of reference problematic. Likewise, the idea of the "total energy of the universe" is difficult to define properly. Misner, Thorne and Wheeler (one of the standard texts on GR) discuss this at length in chapter 20 of their book.
Another approach is Wald's "Hamiltonian" or "Hamilton function" for GR as derived in his GR text. In classical physics, this function can (almost always) be interpreted as representing the total energy of a given system. Using this formalism, Wald shows that, for a closed universe, the Hamiltonian is zero. Similar arguments can be applied to the same effect for a flat universe, although for an open universe the formulation for the Hamiltonian ends up ill-defined.
Other efforts to deal with conservation of energy in GR have used so-called "pseudo-tensors". This approach was tried by Einstein, among many others. However, the current view is that proper physical models should be formulated using only tensors (see again Misner, Thorne and Wheeler, chapter 20), so this approach has fallen out of favor.
However, this leaves us with something of a quandary: in the absence of a proper definition of gravitational potential energy, the law of conservation of energy from classical mechanics clearly does not hold in GR. Thus, for any theory based on GR, like BBT, conservation of energy is clearly not something that can be held against it. Hence, the first law of thermodynamics argument becomes moot.
