Here is my understanding of the central limit theorem: The arithmetic sum/mean of a sufficiently large number of iterates of independent random variables approaches a normal distribution.
These variables should be independent.
I can visualize with a few examples how as a chance process (analogous to drawing from a box) is repeated, the probability histogram centered at the expected sum with standard error as a SD starts to resemble a normal curve, whether or not the contents of the box originally do resemble that.
However, why is this so? I don't even know where to start.