The Laplacian operator is a template which implements second-order differencing. (zero-crossing edge detector)
f''(x) = f'(x) - f'(x+1)
= -f(x) + 2f(x+1) - f(x+2)
The 2nd order template horizontal / vertical looks like
-1 2 -1
0 -1 0
-1 4 -1
0 -1 0
it is important to ensure that the sum of template coefficients is zero, so that edges are not detected in areas of uniform brightness.
Gaussian smoothing first, then Laplacian. Since the convolution operation is associative, we can convolve the Gaussian smoothing filter with the Laplacian filter first of all, and then convolve this hybrid filter with the image to achieve the required result (LoG).