Fibonacci numbers have become a popular introduction to recursion for Computer Science students and there\'s a strong argument that they persist within nature. For these reasons
I don't think there's a definitive answer but one possibility is that the operation of dividing a set S into two partitions S1 and S2 one of which is then divided into to sub-partitions S11 and S12, one of which has the same size as S2 - is a likely approach to many algorithms and that can be sometimes numerically described as a Fibonacci sequence.
Just to add a trivia about this, Fibonacci numbers describe the breading of rabbits. You start with (1, 1), two rabbits, and then their population grows exponentially .
The Fibonacci numbers have all sorts of really nice mathematical properties that make them excellent in computer science. Here's a few:
I'm sure there are more reasons than just this, but I'm sure that some of these reasons are the main factors. Hope this helps!