A Calabi-Yau manifold of genus 0 is topologically a sphere of some kind. Given that, how or why would we expect there to be so many rational curves in that space? Why is the total number of rational curves (of higher order) discrete, and why are there so many of them?
I have also heard a rumor that mirror symmetry gave some mathematicians in Scandinavia a computer method for calculating the total uniquely-identified rational curves.
I think I just need a good diagram to see what is happening here. I think to help me pin this down, give me a specific example of a genus 0 Calabi-Yau, and then laboriously list all of its rational curves. Then tell me why there are only that many and why those are 'unique'.