It's a fascinating ratio, not just for describing spirals. You can split a line into the Golden Section, or make a Golden Rectangle. The classic 1.66 aspect ratio is close to a Golden Rectangle (1:1.62). It occurs in nature because it's related to growth, maintaining the same proportions while expanding. The Fibonacci series, in which each number is the addition of the previous two, generates the ratio between successive numbers, becoming closer as the series progresses - ie 3/5, 5/8, 8/13, 13/21.. I suspect we find it pleasing because we intuitively recognize the growth pattern within it.

Artists and architects have deliberately used it for centuries, but you naturally arrive at it by splitting something into two parts where the small part has the same proportion to the large part as the large part has to the whole. Dividing something in half creates a static or formal sense of space, in thirds it becomes more dynamic but still evenly distributed. The Golden Section splits something in such a way that it is neither static nor even, but retains a hidden proportionality. It's pretty clear most artists use these ways of dividing space intuitively, but it's interesting to analyse them mathematically.