(I'm a nooby in probability)
So why are IQ test results normally distributed? Or more precisely what are the hypothesizes and theorems that imply this distribution?
Has it to do with the central limit theorem? (But this theorem is about the arithmetic mean of iid variables. I dont see iid variables here: I suppose it's not one person repeating the test. Is it the skills given at a person that is considered as a random variable?)
It has been an empirically observed fact that many "naturally" observed traits, like height or IQ, are NOT empirically normally distributed. At the very least they can't be truly normally distributed because they are always non-negative. But even more than that, before non-negativity is violated, it has been observed that the "tails" (values enough standard deviations away from the mean) tend to have higher probability than predicted by a normal distribution for the population, at least for certain traits. The only thing you can say is that if you take many samples and compute the mean, then the empirical mean for the sample should be approximately normally distributed under mild assumptions if you have enough samples (this is the central limit theorem).
As an aside, if you'd like a speculative theory for why many traits appear "somewhat normal", just consider the possibility that many factors affect the trait, e.g. many genetic factors and many environmental factors. If you have many factors and their effects are additive and you don't have too crazy distributions for each factor's effect, and the factors are independent enough, then the accumulated effect should be somewhat normal basically by the central limit theorem.