Quantum Field theory question just conceptually grasping from Sean Carroll's “Biggest Ideas in the Universe”!

Quantum Field Theory from Sean Carroll’s Biggest Ideas in the Universe. I’m just checking to see if I’m on the right track of what he’s explaining. He talks about a free field (non-interacting field), we then get $Psi[phi(x)]$ is the complex amplitude of the field configuration throughout the space. Take the magnitude of $Psi$ you get the probability of observing your field to be in the state of $phi(x)$ correct?

He then goes onto saying we can treat this as particles. We take our $Psi$ Function and can use Fourier analysis to break it up into "modes" which he calls simple harmonic oscillators, the energy is proportional to $h^2$ where $h$ is the height of the wave. Then since these energy states have discrete states at equal intervals then we can treat them as $n$ number of particles, since there’s $n$ number of energy states.

Am I understanding this correctly? Is there anything big in intuition gap that I’m missing? I know this is for a specific field type, there’s others like vector, tensor and so forth but he saves those for later I believe. I don’t have too much physics background, I was a math major at UC Santa Cruz and had to drop out at senior level, so I do have a fair share of math if you need to explain it that way! Thank you and sorry if the question seems vague. I’m doing this for fun and trying to learn more on my own.