The string post relies on its curved shape to allow the string tension to push the string upwards so that the end through the hole is pressed against the top of the hole by the string wraps beneath it. Too few turns, and the string just wraps around the middle of the post without the extra grip of the string being pushed against the top of the hole.

The downside of needing several turns is that when you use there trem, the string tension is reduced significantly and the wound strings especially can move a bit relative to one another, resulting in a slightly different string tension when the trem is released, and so the string goes out of tune until any slack in the windings is taken up again whilst playing. A smooth, well-cut nut will minimise this as the string will slide easily through the nut slot without any tendency to snag. But if you do use the trem a lot and find problems, then swapping to locking tuners can help, as you shouldn't ever have a full turn of string around the post before it's in tune, so there's no string against string interaction/friction.

If you don't have a trem or don't really use it, then they just make string changing quicker.