As btilly points out elsewhere, to actually obtain the correct limit, you have to normalize the sum correctly. Because of the way the scaling is done in this graphic, as you increase the number of levels, it's in effect normalizing the sum by dividing by "n". To get the right limit, you need to divide by sqrt(n).

In this sense, the CLT is a high-resolution version of the SLLN ("law of averages"). If you normalize the sums by 1/n, the resulting average converges to a number, the mean.

But if instead you subtract this mean, and normalize by 1/sqrt(n) rather than 1/n, the result converges in distribution, and looks like a normal random variable.

By less aggressive normalization, you get information about the fluctuations rather than just pounding it down to a number, as the SLLN does.

I used to TA a probability class for undergrads, and I found the demo to be perfectly reasonable.

The reason I posted is to defend the OP against some assertions that I felt were picking at details of what started out to be a simple and fun visualization. I wrote what I did about the LLN to supply some intuition to the person who created the visualization, in case they wanted to put these two results in perspective.

  var( (1/sqrt(n)) * (X_1 + X_2 + X_3 + ... X_n) 
    = (1/n) * (var(X_1) + var(X_2) + ... var(X_n))
    = (1/n) * n * var(X_1)
    = var(X_1)

The CLT tells us more than that, it actually tells us how the randomness is distributed when n gets very large, which is pretty remarkable when you think about it. (and it holds under much weaker conditions than what I mentioned above, it's just that those assumptions are probably the easiest to understand).