How do developing neurons become tuned to fire only on receiving specific inputs?
The only "good model" I know of is hebbian plasticity* or hebbian theory, here's a link: https://en.wikipedia.org/wiki/Hebbian_theory

Usually what we mean by neurons "listen for" or are "tuned for" or even "like" a particular stimulus is that spiking activity in those neurons is significantly increased (or decreased) in the presence of that stimulus as compared to equal amount of time in the absence of it.

*Plasticity refers to the capacity of the brain to change both the strength of connection as well as number and configuration of connections.