Gaussian filters are used in image processing because they have a property that their support in the time domain, is equal to their support in the frequency domain. This comes about from the Gaussian being its own Fourier Transform.
What are the implications of this? Well, if the support of the filter is the same in either domain, that means that the ratio of both supports is 1. As it turns out, this means that Gaussian filters have the 'minimum time-bandwidth product'.
So what you might say? Well, in image processing, one very important task is to remove white noise, all the while maintaining salient edges. This can be a contradictory task - white noise exists at all frequencies equally, while edges exist in the high frequency range. (Sudden changes in spatial signals). In traditional noise removal via filtering, a signal is low pass filtered, which means that high frequency components in your signal are completely removed.
But if images have edges as high frequency components, traditional LPF'ing will also remove them, and visually, this manifests itself as the edges becoming more 'smudged'.
How then, to remove noise, but also preserve high frequency edges? Enter the Gaussian kernel. Since the Fourier Transform of a Gaussian is also a Gaussian, the Gaussian filter does not have a sharp cutoff at some pass band frequency beyond which all higher frequencies are removed. Instead, it has a graceful and natural tail that becomes ever lower as the frequency increases. This means that it will act as a low pass filter, but also allow in higher frequency components commensurate with how quickly its tail decays. (On the other hand, a LPF will have a higher time bandwidth product, because its support in the F-domain is not nearly as big as that of a Gaussians').
This then allows one to attain the best of both worlds - noise removal, plus edge preservation.