How to understand Kalman gain intuitively?
I found a good way of thinking intuitively of Kalman Gain $K$. If you write $K$ this way

$\displaystyle \quad\ \bf{K_k} = \bf{P_k^-\, H_k^{\rm T} (H_k P_k^-\, H_k^{\rm T} + R_k)^{-1}}
  = \bf{\frac {P_k^-\, H_k^{\rm T}}{H_k P_k^-\, H_k^{\rm T} + R_k}}$

you will realize that the relative magnitudes of matrices ($R_k$) and ($P_k$) control a relation between the filter's use of predicted state estimate ($x_{k}⁻$) and measurement ($ỹ_k$).

$\displaystyle \quad\
  \lim\limits_{\bf{R_k \to 0}} \bf{{P_k^-\, H_k^{\rm T}} \over\
  {H_k P_k^-\, H_k^{\rm T} + R_k}}\
    = \bf{H_k^{-1}}$

$\displaystyle \quad\
  \lim\limits_{\bf{P_k \to 0}} \bf{{P_k^-\, H_k^{\rm T}} \over\
  {H_k P_k^-\, H_k^{\rm T} + R_k}}\
   = \bf 0$

Substituting the first limit into the measurement update equation 

$\displaystyle \quad\
  \bf{\hat x_k} = \bf{x_k^-} + \bf{K_k}(\bf{\tilde y_k}-\bf{H_k}\bf{x_k^-})$

suggests that when the magnitude of $R$ is small, meaning that the measurements are accurate, the state estimate depends mostly on the measurements.

When the state is known accurately, then $H P^⁻ H^T$ is small compared to $R$, and the filter mostly ignores the measurements relying instead on the prediction derived from the previous state ($x_k⁻$).