You already have several excellent answers, but from your question I suspect you are hoping for an answer that is a bit less mathematical and more about just how the transition between quantum physics and ordinary large-scale physics takes place.
For most everyday phenomena, including chemistry and the way behaves and interacts with matter light (radio, light, X-rays, gamma rays), quantum mechanics already predicts classical phenomena with exquisite precision, as best we can tell.
The "as best we can tell" qualifier is because the computational cost of doing so is horrendous and grows larger so quickly that you have to start approximating very early in the process. The theory that provides this level of precision -- arguably the most precise predictive theory ever developed -- is something with the ungainly moniker of Quantum Electrodynamics, or QED for short. The variant of QED that Richard Feynman co-received a Nobel Prize for is the one that led to Feynman diagrams, those little stick-figure diagrams that show electrons as arrows and light particles (photons) as squiggles.
QED also shows beautifully how such odd physics can lead to our ordinary world: Through probabilities. Many very strange things are possible in quantum mechanics, including for example light moving around in random loops and turns instead of in a straight line. QED even allows you to calculate the probability of such things, and then build experiments to very that such oddities really do exist!
But what happens on the scale of ordinary humans is that these oddly quantum scenarios quickly become so incredibly minute that they simply don't happen, at least not within the life span of the universe. So light, for example, seems to go in straight lines for the most part (even on our scale that's not entirely true) because the probabilities for all those odd paths, or even paths slightly off from straight, become vanishingly small.
Added all together, this collection of "reasonable probabilities" for many, many atoms and particles of light becomes what we think of as ordinary or classical physics. Surprisingly, the transition between the two via probabilities is quite smooth and not abrupt in any way. We just don't notice that transition much because, except for certain large-scale phenomena such as metallic mirrors that seem "classical" only because we have simple rules to approximate how they work, all of these transitions from strange quantum physics to the comparative simplicity of classical physics take place at very small scales, typically near the size and mass scale of atoms and their constituent particles.
QCD: Going Nuclear
For nuclear phenomena, a similar theory that by intentional analogy called QCD (for Quantum Chromodynamics) does a pretty good job of predicting why particles like protons and neutrons -- and many other less common particles -- behave like they do. That theory is even more difficult to compute than QED, however.
The Standard Model
Beyond that is the Standard Model, an intensely quantum-based model that ties together the entire zoo of particles that we see coming out of particle colliders. Although the Standard Model addresses only a very narrow and exotic range of predictions of phenomena that we never encounter directly in everyday physics, it nonetheless is critical for explaining much of how the large-scale structure of the universe emerges. Through this roles in defining the universe in which we exist, the Standard Model also helps explain how quantum phenomena lead (much less directly!) to what we call the classical world.
So what's missing? Gravity!
Gravity remains ornery and uncooperative in terms of its quantum description providing precise predictability. That's not from any lack of trying! Speculations on quantum gravity are in fact the darling of many popular programs and ideas about physics. But because the original theory of general relativity by a fellow named Einstein was a purely geometric theory without a shred of quantum anything to it, it remains to this day a theory that is difficult to fold into the kind of pure-quantum framework exemplified by the spectacular success of QED. That might not be a problem if general relativity was an inaccurate or approximate theory, but that's not the case: Like QED for its domain of electrons and light, general relativity for its domain of the overall structure of a universe with gravity remains spectacularly effective and predictive for what we can see.
So, for much of the world, and all of the parts of it that we see and interact with on a daily basis, the transition between quantum physics and ordinary physics is already surprisingly well understood from a mathematical perspective, even if philosophical views are far from being in agreement: It's all just a matter of probabilities, with very small things allowing more (and more strange) things to go on, and classical physics just being the sum of probabilities that begin to get very specific and very selective at larger scales. There are holes still, sure, but it seems likely that even when those holes are someday filled, that theme of higher probabilities providing the transition will remain.