btilly gives a starting point for a derivation, which is great. Here are two other ideas, working from the other direction. Notice that the Gaussian is closed

btilly gives a starting point for a derivation, which is great. Here are two other ideas, working from the other direction. Notice that the Gaussian is closed under rescaling, that is, if the variables X1...Xn are all iid Gaussian, then the properly normalized sum is N(0,1). In this sense, the Gaussian is an attractor of the normalized transform. Another worthwhile experiment is to generate 6 or 10 iid uniform variates, and notice how close the normalized sum is to Gaussian. It converges rather fast. You can do this by Monte Carlo, or symbolically using Mathematica.