Don't overthink things. Precal is not deep.
The fact that i^(2)=-1 IS what imaginary numbers are. You write them down, and FOIL things like normal. Understanding IS memorizing their properties and connections. You get intuition by doing problems well. That is a kind of knowledge that you have to read. It's not something divine or philosophical.
Similarly, usually the "why" is "that's the definition" or "because that's what it's for." The reason you use the quadratic formula to solve quadratic equations is because the quadratic formula solves quadratic equations. It's really more about reading instructions and knowing what you want to accomplish, not some deep why.
If you want to understand more of that, make sure you know your theorems. For example, why do we factor polynomials in equations like polynomial=0? It's because there's a theorem that says if you have AB=0, then you know A=0 or B=0. How do you know to apply it? Because the theorem starts with "AB=0", and you have a thing=0.
If you really want something to build an intuition around, I say focus that energy on visual stuff. For example, it's good to practice a lot of problems involving graph shifting/stretching. Because visualization is actually skill (although incidentally it involves just memorizing some graphs), unlike calculations which is just following instructions.
