In short: Classical Physics is the limit to infinity of Quanta in Quantum Mechanics. Thats in a nutshell.
For people who are not familiar with the mathematical concept of limits consider this example:
Say, F(X)=1-5*10^-X (one minus five times ten to the power of minus X). In this example:
This is the concept of a limit, notice that for small values of X, say between 0 and 3 the value of the function changes dramatically. But from the values of 4 and above the change is negligible. Consider how much F(4) is close to one!!
So in Quantum Mechanics the formulas are more complex than Classical Physics, but when the number of Quanta is large, certain parts of the equation can be discarded since they are negligible. Just like how in "F(X)=1-5*10^-X" we can discard the "-5*10^-X" since its almost zero for large values of X. The concept of limits apply when moving from Quantum Mechanics to Classical Physics.
Another hot topic is the probabilistic model of Quantum Mechanics. While we cannot be certain of the outcome of Quanta in overly simplistic scenarios, this does not apply to large numbers of Quanta. Its the same concept of limits.
Consider the following scenario:
Say X is a variable that initially has the value of 0. We throw a die, and we add the value of the die to X. So lets see what happens to X after 3 throws.
Say, the outcome of the die was: 3 / 3 / 6
Another experiment has this outcome: 4 / 1 / 2
Yet a third with this outcome: 5 / 6 / 4
It seems that X is different for each of these trials... This is an analogy to what happens with Quanta. Each Quanta produces a different stochastic (random) behavior. And the behavior of a system of Quanta is the sum of all the behavior of Quanta at that moment.
Consider now the above example with a billion throws. While the outcome of 3 trials might have large differences in each experiment, as the number of trials increases the error margin decreases. The the most basic property of probability. The more the trials the more stable results you will get.
In the above example, I can say that the average outcome of the die is (1+2+3+4+5+6)/6 = 3.5
So in one billion trials, the outcome should be very close to 3.5 billions.
The conclusion is that although when we consider Quanta in isolation they might produce diverse results, studying large numbers of Quanta is much easier and easily predictable with high confidence.
It is worth noting that one proton contains: 2.26*10^23 Quanta!! As you can see, even a single subatomic part contains a huge number of Quanta!! So even for a single atom assuming that the number of Quanta in it is infinite is a reasonable approximation in limit equations!