Do we really not know why atoms 'decide' to produce a photon? I was watching the Cosmos documentary where Neil deGrasse Tyson explained how certain energy photons get absorbed by an atom, which causes the electrons of that atom to climb into a higher energy state.
He then says that an atom produces a photon when those electrons drop to a lower energy state, but that we don't know why this happens. I understood that as 'we don't know what triggers this to happen'.
Is that true? And if so, are there any feasible theories that explain this phenomenon?
 A: 
He then says that an atom produces a photon when those electrons drop to a lower energy state, but that we don't know why this happens. I understood that as 'we don't know what triggers this to happen'.

As others have said, for an individual atom with an electron in an excited level,from the theory of quantum mechanics one can only give the probability that the electron will fall back to the lower energy level for this individual atom under observation. The probability distribution accumulated for all the photons from atoms in the same excited state is predicted with great accuracy by the theory.
"Why" questions in physics end up on the axioms and postulates of the theory. Physics answers "how questions", how from postulates and axioms one can  describe the observations with the model, the Schrodinger solution of the hydrogen atom,for example. "Why" questions end up in "because this is what we observe".
In this particular case the "why" is answered with "because energy falls to the lowest possible level". The same is true with balls falling by the attraction of gravity, except the energies involved are not quantized.
Our mathematical physics theories are based on  observed conservation laws, and energy conservation is one of the basic ones. The theories that describe the data have attractive and repulsive forces , and attractive forces change potential energy to kinetic energy classically, as with the ball falling in the gravitationalpotential. In the atom's case the attractive force between electrons and nuclei   changes potential energy in  quantized steps, if a lower energy level is available, and the energy leaves as a photon. This is an observation fitted extremely well by the theoretical models.
A: Yes, in the sense that you understand the "Why does this happen?", we really don't have an answer.
That an electron emits a photon is an allowed interaction in the underlying quantum (field) theory. This process has a certain probability to occur. And that's all we can say about it. As far as we know, there is no "trigger" for the emission, it is truly a random process occuring with a given probability.
To hope that there is a theory that does away with this kind of probabilism is to hope that there is a theory of hidden variables. Indeed, at least one interpretation of quantum mechanics, the Bohmian, is such a (nonlocal) hidden variable theory that would deterministically predict when and what happens if we knew the initial state of our system perfectly - this theory explains the observed probabilism then by our ignorance of the system, so that its predictions do not differ from a "truly probabilistic" interpretation. 
Furthermore, Bell's theorem states that any theory that agrees with quantum mechanical predictions is either non-local, or has no unique predetermined measurement results (isn't realist). This means that you cannot ever get a theory that answers our questions of "Why?" as we wishes it did, because every theory that predicts unique results violates the idea that stuff can only influence each other at the speed of light, and every theory that plays nice with our relativistic idea of causality has no predetermined measurement results to speak of.
Therefore, it is indeed true: We have no idea "why" the electron emits a photon, and it is highly unlikely we will ever be able to say more than that it simply does. (This should not be surprising: We all know that "Why?" is an annoying, infinitely repeatable question. We also know that human knowledge is necessarily finite, so there will always be some point at which we cannot answer the why, and it is indistinguishable whether that is because we haven't found out yet or because there simply is no answer.)
