[nine_k]

Fields are continuous. E.g. electromagnetic field in the classic electrodynamics is continuous. But its waves, photons, show quantum, discrete properties — see the photo effect. It would be easy if there were some common grid, but it's not the case. You can make photons with whatever energy from a continuous spectrum, but photons of each frequency will carry its own quantum (Latin for "amount") of energy, in discrete portions.

Why is this happening? That's the question worth 1000 Nobel prizes.

[GoblinSlayer]

It's called second quantization and is taught in the course of theoretical physics. https://en.wikipedia.org/wiki/Second_quantization

[nine_k]

This is the how, not the why. That is, it's not a (more general) model from which quantization follows. Obtaining such a model would be a major breakthrough, provided that it's falsifiable in humanly-attainable conditions (unlike the string theory).