I saw a pretty fun series of videos explaining why this is.

Viewing Fibonacci spiral structures in plants shows that as the structure grows from the meristem of the plant, each new growth is offset by the golden angle from the previous growth, rotating in a manner that many revolutions will show spiral formations in Fibonacci numbers.

French physicists Douady and Couder performed some experiments by periodically dropping magnetic fluid onto some oil and observing how they moved from that center point. Drop them slowly enough and they'd alternate sides, by repelling each other. But drop them close enough to where the repulsion happened against multiple droplets and they'd tend to stabilize at rotating around, with the golden angle.

So the simple algorithm of growing away from the previous growth is easy to encode genetically, and creates spiral structures with minimal overlap with prior growths. So Fibonacci spirals arise with the simplest of growth instructions coded into the genetics.