How does the happiness affect the population in Stellaris?

This answer is for version 2.2 "Le Guin". The main impact of pop happiness is its effect on planetary stability through a mechanic called approval rating. The approval rating of the population on a planet is a function of their happiness and their political power: in a stratified society, pops in lower strata don't matter as much as those in higher strata.

Stability has some effects that are similar to the old effects of pop happiness: high stability gives job production bonuses, low stability production penalties. High stability also increases Trade Value. High stability adds immigration to the planet (increasing pop growth) while low stability adds emigration (slowing pop growth, or making pops decay if it grows too high - this is partially good, because overpopulation is a major stability penalty). Finally, if your stability is low, it can lead to events such as unrest, riots, etc.

Compared to the previous system, the important thing to note is that stability is planet-wide. If you have a mix of happy and unhappy pops, the happy pops will suffer a production penalty as well. However, you can reduce that by having more enforcer (police) jobs on the planet.

Compared to the system at the time of the previous answer, there are no longer happiness breakpoints - that is, every increase in happiness will result in an increase in production (unless you're running up near 100%), without requiring you to hunt for specific percentages. Also, habitability is now a penalty, rather than a cap, to pop happiness. If your pop is on a 60% habitability world, it can have more than 60% happiness, but it will have to overcome a ~10% penalty to get there.