I've been in the process of writing a tutorial on how PID filters work for a much younger audience. As a result, I've been looking back at the original tutorials that made stuff click for me. I had several engineers try to explain PID control to me over the course of about a year, but I don't think I really got it until I ended up watching Terry Davis (yeah, the TempleOS guy) show off how to use PID control in SimStructure using a hovering rocket as an example.
The way he built the concept up was to take each component and build on the control system until he had something that worked. He started off with a simple proportional controller that ended up having a steady state error with the rocket hovering beneath the target height. Once he had that and pointed out the steady state error, he implemented the integral term showed off how it resulted in overshoot. Once that was working, he implemented the derivative control to back the overshoot off until he had something that settled pretty quickly.
I'm not sure how you could do something similar for a Kalman Filter, but I did find it genuinely constructive to see the thought process behind adding each component of the equation.