As a completely off-the-wall guess, I would say that since they show up in the branching of trees (and likely the branching of anything, including neuronal axons), and trees are 3-dimensional, then first off, the sequence likely depends on the number of dimensions of the system. The optimal sequence for growth/branching in 2D is perhaps, then, a different sequence to that of our fibonacci sequence.

Given that, it likely says something about the rate of growth/branching in 3D such that optimal space is taken up in 3 dimensions. If a tree branched more rapidly, it would at its extreme grow a shell of branches at its boundary, if it branched more slowly, the branches would thin out and leave it bare-looking. It's the "just-right", middle ground that we got where seeming equal amount of empty space exists throughout the space taken up by the tree including all branches, stems and leaves. I have no evidence for anything I just said