IQ isn't normal, it's normalized to have mean 100 and standard deviation 15, usually via a percentile method.
The reason IQ looks roughly normal is because intelligence (however it is defined) is a complex trait. Complex traits are predicted to have a roughly normal distribution based on the central limit theorem: a sum of many individual factors (including genetic and environmental ones) will tend to be distributed normally in a population, even if the underlying factors themselves are not normal.
There is no real concrete measure "IQ": it isn't measuring a real-world physical property the way you measure mass or length. Instead, you use tests intended to get some measure of that abstract trait, and then normalize individuals based on the group statistics. Actual tests administered to measure IQ will have a minimum and maximum score: you can at worst get every question wrong, at best get every question right.