A Kalman filter is a specific instantiation of the Bayes filter, which is very elegant and intuitive. Then once you plug in a bunch of Gaussians and linearity in the Bayes filter and crunch the math, a Kalman filter falls out.

This is how it's treated in the book Probabilistic Robotics by Thrun, Burgard, Fox and I found it to be one of the best treatments on Kalman filters. A really great book overall.