The best presentation that I've ever seen on what a Kalman filter really is comes from a SIAM Review article, "A Fresh Look at the Kalman Filter" by Jeffrey Hum

The best presentation that I've ever seen on what a Kalman filter really is comes from a SIAM Review article, "A Fresh Look at the Kalman Filter" by Jeffrey Humpherys, Preston Redd, and Jeremy West http://epubs.siam.org/doi/abs/10.1137/100799666 http://epubs.siam.org/doi/abs/10.1137/100799666 It sets up the discrete-time linear system and then uses a minimization principle to show what's going on. I can highly recommend it especially for people coming to Kalman filters from a math or optimization background.