If you want to be someone that understands it very deeply get REALLY good at linear algebra and REALLY good understanding of multi-variate calculus.

The not so deep answer to your questions is your understanding right now is right. You have a bunch of functions that take multiple inputs and spit out 1 output and that output is combined with other outputs to be put into other functions. Each function has parameters that can vary which changes the output. When you train you give a bunch of examples that in real life you know (hope) are related. The model learns parameters such that it maps input to output.

That's all that's happening.