There are two basic forms of sharpening filter:
- I - 𝛼 Laplace(I)
- k I - (k-1) smooth(I)
where I is the image, 𝛼 is a positive value, and k is a value larger than 1.
The first form is referenced by Laurent in his answer, you can use the Laplacian of Gaussian to construct it. The second one is the classical unsharp masking, and was used in photography before computers existed.
Both of these can explain the filter in the OP, as both methods do more or less the same thing. But as you scale them, they become different.
The second form is the easiest to control. It has two parameters: the size of the smoothing filter (and its shape), and the k. The larger k, the stronger the sharpening effect. The larger the smoothing, the stronger the sharpening effect as well, but it also selects for the size of the edges that are sharpened.