Interesting you should ask. An article gives an interesting summary of how they used to do it, but a little blurb at the top gives us the important information:
UPDATE: Unfortunately, this post is no longer accurate with regard to Skype’s infrastructure. After the massive Skype outage in December 2010, it was expected that Skype was exploring ways to make their system more stable and resilient. In early 2012, Skype (at that point now owned by Microsoft) was reported to have replaced much of the P2P supernode infrastructure with supernodes hosted in Microsoft data centers.
So your answer is that they moved it to the cloud because their previous self-run infrastructure proved insufficient. Of course, they probably could have gotten a bunch of money and decided to build their own, and if they had built enough of that, they might have decided to rent out some of that and become a cloud provider in their own right.
Global distributed systems take a lot of work that a lot of people choose not to think about. Me too, when I can get away with it, because a lot of it is boring. I have watched people think they've done it when they really haven't, though, and they tend not to like the results. If you host in a single facility, you're not distributed. If you think that a few colos on different continents does it, you're probably not as interrelated as global communication services are.