An Intuitive Way to Understand the Sharpen Convolution Filter in Image Processing
Consider this filter in 1D (for the sake of simplicity):

[-1 1 0]


This filter responds to pixel differences, i.e. edges (left pixel * (-1) + center pixel * 1 + right pixel * 0) which is (center pixel - left pixel).

Response of this filter is positive on rising edges and negative on falling edges.

The filter is not symmetric and hence it will "shift" the image to the left. To make this filter responding to "edge on the left" and "edge on the right" equally, we have to compensate it by similar filter:

[-1 1 0] + [0 1 -1] = [-1 2 -1]


Now this can produce "edge image". We will enhance the edges by adding them to the original image:

[-1 2 -1] + [0 1 0] = [-1 3 -1]


This kernel have sum of 1 and does not produce negative values. It is the desired sharpening kernel.

Summary: Sharpening =  edge detection + original image.