I think the easiest way depends on your background knowledge. If you understand linearity of the Gaussian distribution and the Bayesian posterior of Gaussians, the Kalman filter is almost trivial.

For (1D) we get the prior from the linear prediction X'1 = X0*a + b, for which mean(X'1) = mean(X0)*a + b and var(X'1) = var(X0)*a^2, where a and b give the assumed dynamics.

The posterior for Gaussians is the precision weighted mean of the prior and the observation: X1 = (1 - K)*X'1 + Y*K, where the weighting K = (1/var(X'1))/(1/var(X'1) + 1/var(Y)), with Y being the Gaussian observation.

Iterating this gives the Kalman filter. Generalizing this to multiple dimensions is straightforward given the linearity of multidimensional Gaussians.

This is how (after I understood it) it makes it really simple to me, but things like linearity of (multidimensional) Gaussians and the posterior of Gaussians as such probably are not.