The mathematical properties of this sequence are easy enough to understand. The marvel comes from the fact that it is instantiated all over the place. You've seen a stunning animation of this phenomenon before, but why don't we put the idea to the test with some real-world examples:
As she said, there's nothing really mystical about the appearance of this number in these plants. If you consider this to be an engineering problem requiring the most efficient packaging solution, then it shouldn't be that surprising that the blind process of evolution should have eventually stumbled upon this pattern. For more, take a listen to this great discussion of the Fibonacci sequence and the golden ratio from In Our Time.
If you're still hungry for more, learn more about fractals from none other than Arthur C. Clarke, or how simplicity can give rise to complexity in The Secret Life of Chaos, with Jim Al-Khalili.
And don't forget to check out parts 2 and 3 of this awesome little series.