Planck's Constant is said to be one of the smallest constants used in Physics. Some even say it is one of the imponderables – something that we know exists and is true but cannot be explained yet. I remember reading somewhere that Max Planck somehow discovered this constant but was unable to explain how he did so. While experts may dismiss it as a concept of quantum mechanics, a branch of science that is still under intense research, a common man might see it as nothing short of a mystery.
Planck just introduced the constant, because when he did so, he suddenly arrived at the law for black body radiation that now bears his name and back then explained the full spectrum of radiation for the first time in history (all other phenomenological laws broke down at some point). Why he introduced it - it was just a long way of trial and error. It just fit, but Planck didn't really believe it to be something physical and neither did the other physicists back then (basically until Einstein and a bunch of others showed how the concept of quantization was the only possibility to explain the black body radiation and also helped explain other phenomena).
So how did Planck arrive there? He tried everything he could think of and then finally made an ansatz involving certain boundary conditions that lead to a quantization of the energies (something that was already known at the time), then he got a curve that he could fit to the data by adjusting a parameter and voilà, everything worked out. Imposing finite boundaries is something frequently done in statistical mechanics (to take the thermodynamic limit later) and was already known at this time - although frowned upon.
As for the other parts of the question: Physics is not really about the "why something is true", it's more about "how does it work" part. That means that ultimately, we have to make at least a few assumptions and probably fix a few parameters, so it's not "experts dismiss[ing] it as a concept of quantum mechanics", but experts accepting that quantization (and hence the need for Planck's constant) makes our theories describe the real world. Maybe this also means that at certain scales our world is quantized, maybe it doesn't. The "mystery" is metaphysics and hence belongs to the realm of philosophy.