There many different ways to discuss this transition. The most straightforward one uses the path integral formulation of quantum mechanics, in which the transition amplitude from some state $\left| \psi \right\rangle$ to another state $\left| \phi \right\rangle$ is found by adding up all of the possible classical trajectories (in phase space) that could in principle interpolate between these two states, each one weighed by $\exp(iS/\hbar)$, where $S$ is the action for the trajectory in question. \begin{equation} \left\langle \phi,t_f|\psi,t_i \right\rangle = \sum_{\text{all histories}} e^{iS/\hbar}. \end{equation} For a macroscopic system, the action for most histories is an enormous number in units of $\hbar$, whose value changes appreciably even between phase space paths which are very similar. The result is that the complex exponentials tend to cancel each other for most of the paths in phase space, due to the rapid oscillations causing a sort of ''destructive interference'' in the sum. The only exceptions to this are the histories which happen to sit in a minimum of the action. For these, the action will have a lower value that does not change so rapidly between neighboring phase space trajectories around this minimum, so their phases do actually add constructively. The end result is that for macroscopic systems, the transition amplitude is dominated by the contribution of the histories that lie on the minima of the action, which are the classical trajectories. Most graduate level QM books discuss this in detail. One undergraduate book that has a nice explanation of this is A Modern Approach to Quantum Mechanics, by Townsend.

Another, completely different way of looking at how a quantum system becomes classical is by thinking in terms of decoherence. This a more complicated subject, which explains how small perturbations to a quantum system due to interactions with the environment end up destroying its quantum properties and causing it to ''collapse'' (or decohere) to its classical configurations. Since it is impossible to isolate a macroscopic system from the environment, this is a way to understand why such a system effectively obeys classical mechanics, despite its fundamental quantum nature. An excellent pedagogical introduction to this is https://arxiv.org/pdf/1508.04101.pdf, in which this phenomenon is worked out explicitly for a toy model. The original papers by Zurek on decoherence are also recommended, as well as his talks, which you can find on Youtube. Here is one of them https://www.youtube.com/watch?v=7Sn63t3BeMc.