It is the distribution of the average of n binomial distributions (taking on values x±1)†, which according to the CLT converges to a Gaussian as n→∞.
[†] x is where it is centered.
[†] x is where it is centered.
In the visualization as you increase the number of bins, the Gaussian approximation becomes more and more squeezed, and by the weak law of large numbers the limit is a Dirac delta.
In order to get the Gaussian you'd need to be looking at a window whose width is proportional to the square root of the number of bins.