Help Understanding Convolution Filter Formulation
Te elaborate on nikie's comment that it is just a change of variables, consider the following.

Suppose that the original image is a single pixel of intensity $1$ at point $(0,0)$
and all other pixels have intensity $0$. Then,
the result of the convolution of this image is $h(i,j)$, that is, this
single pixel in the original image produces pixel intensity $h(i,j)$ at
the point $(i,j)$.  The actual image is a whole bunch of pixels, though, at
different points, and the superposition principle says that the resulting
pixel intensity $g(i,j)$ at point $(i,j)$ is the sum of of the pixel intensities
produced by the various pixels in the original image. 
There are two ways of calculating $g(i,j)$.
Let's go through the pixels in the original image in order of their
distance from the point $(i,j)$.  Well, the point $(i-k,j-l)$ is at distance
$(k,l)$ from $(i,j)$ and the pixel intensity $f(i-k,j-l)$ there
produces intensity $f(i-k,j-l)h(k,l)$ at $(i,j)$.  This gives the first
formula
$$g(i,j) = \sum_k\sum_l f(i-k,j-l)h(k,l).$$
Alternatively, we can go through the pixels in natural order and ask
what intensity the pixel $f(k,l)$ produces at the point $(i,j)$. Well,
the distance is $(i-k,j-l)$ and so the intensity is $f(k,l)h(i-k,j-l)$
giving
$$g(i,j) = \sum_k\sum_l f(k,l)h(i-k,j-l).$$