Why Does the Kalman Filter Remove Only Gaussian Noise?
A Gaussian process is completely specified by its mean and variance.  A Kalman Filter updates the process mean which is the state, and its variance.  These are the sufficient statistics.  

The measurement noise is reduced but the process noise  is part of the recursive state history and is tracked. 

Your heading question and subsequent paragraph aren't fully consistent.  One can use a linear Kalman Filter when the noise isn't Gaussian in many circumstances but it wouldn' be optimal.  The derivation assumes Gaussian (or Normal) noise and all deterministic inputs are known, as well as knowing initial values.