Not a lot of ELI5 answers, but some good history.
The Fibonacci sequence is a set of numbers with a distinct pattern (explained in other comments). What is important is that the ratio of one number to the one following it is always the same. (The second is always 1.618 times larger than the previous). That is called the golden ratio, and it is the golden ratio that is seen everywhere in nature.
If you’ve seen the image of rectangles that form into a spiral, this is what it means:
The small rectangle has sides with that exact ratio. The long side of that rectangle is the short side of the next, and that rectangle uses the golden ratio. The long side of that one is the short side of the next…. And so on. This creates a spiral pattern, and that pattern, in that ratio, happens all the time. Flowers, tree leaves, and animal shells for example. Always 1.618 times bigger than the previous part.
The number isn’t magical. 1.618 isn’t special. There is just a natural order to things, and we created a numerical system that happens to measure that order at that number.
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