Ladybird vs. Aphid.... vs. Spider?

I thought Ladybirds ate Aphids and kept their population down. But apparently that isn't the whole truth! Check this out:



I don't know about you but I really do wonder if this is at all true. The roles of both the spider and the Ladybird seem so passive. I reckon there's something missing in the equation.

http://www.bbc.co.uk/nature/life/Coccinella_septempunctata#p0039ryj