Function for response over image scales
Conceptually this works as follows: first you construct a Gaussian scale space by convolving the given image with the Gaussian filters of increasing $\sigma$. This is where the sigma comes from.  Thus you have a 3D scale volume, where each plane is an increasingly blurred version of the original image.  Then you evaluate your function $F$, which in your case would be a Laplacian, over each scale plane, i. e. at each $\sigma$.  Then you find local maxima of $F$ over $x, y$, and $\sigma$.

Note that in practice you can use the fact that convolution is a linear operation in various ways.  For example, you can generate each scale plane from the previous plane, rather than from the original image.  This allows you to use the same filter, rather using several filters with increasing $\sigma$.  Or you can generate $F(\sigma)$ directly from the original image by using the Laplacian-of-Gaussian filter of the corresponding $\sigma$