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A Kalman filter is designed to minimize errors in a linear system. However, it can be applied to non-linear systems by assuming that small changes in the system are linear. The estimated system state is (hopefully) close to the actual state, so this may be a reasonable assumption.
The matrix of Jacobian derivatives is simply a way of taking the non-linear system and making it linear, by off-setting the state to the current estimate and using the the derivatives of the predict and update functions. The earlier assumption is that the derivatives are constant for small errors in the state, so then the Kalman filter can be used. Note that the Jacobian has to be reevaluated at each filter point.
This method is called the Extended Kalman filter. It is useful if the functions are easily differentiable and not overly non-linear.
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