Why does the bell curve turn up in so many different places?

Because sums converge on it. The central limit theorem states that the normalized sum of independent random variates with finite variances approaches a normal distribution. Each variate may have an arbitrary probability distribution of its own; what matters is that they are independent and that their variances are finite.

The central limit theorem states that the normalized sum of independent random variates with finite variances approaches a normal distribution. Take a set of independent random variates, each with an arbitrary probability distribution, its own mean and a finite variance.

The normal form variate built from them has a limiting cumulative distribution function that approaches a normal distribution. Under additional conditions on the distribution of the addend, the probability density itself is also normal.

The individual shapes do not matter. That is the whole force of the theorem: heights, errors and marks arrive from unrelated processes, and the sums behave the same way regardless.

Proofs come in several lengths. Kallenberg gives one in six lines, while the elementary but more cumbersome route works through the inverse Fourier transform, expands, and recognizes the result as the Fourier transform of a Gaussian function.