I know that gradient formula in other coordinate system is more complicated than the Cartesian formula. For example, the gradient in polar coordinate system is
$$ \nabla f = \frac{\partial f}{\partial r} \hat{\vec{r}} + \frac{1}{r}\frac{\partial f}{\partial \theta} \hat{\vec{\theta}} $$
I can follow the proof of this result (for example here), however I don't understand the result intuitively. Why is the result different from that in Cartesian coordinates? Can somebody shed some light on it?
When you change coordinates your vectors are naturally expressed in the 'local' basis of the new system. These are vectors tangent to the coordinate lines of the new system (for ex. Circles and rays emanating from the origin in the case of polar coordinates). The basis changes from point to point and because of that the coordinates of vectors change as well since the vector itself must remain invariant. It is like measuring something using different units you get different values to compensate for the difference of units. This philosophy applies to general tensor fields. By the way the gradient is not really a vector but a covector which transforms as a linear form, the differential of the function.
