What is the reason of using Gaussian model for Background Modeling?
The Gaussian function is ubiquitous in statistics and many other fields. One reason is the Central Limit Theorem, which says that the means of independent identically distributed random variables themselves have a normal (Gaussian) distribution. This is why you see the bell curve in nature so often.

The Gaussian PDF may not look simple at a first glance, but it is actually very simple to differentiate and integrate. Remember that exp(x) is its own derivative and its own indefinite integral. It is also one of very few functions whose Fourier Transform can be computed analytically.

Finally, the reason Stauffer and Grimson and others used the Gaussian Mixture Model, is that it is an easy way of approximating a complicated multi-modal distribution.