Umm... maybe i am misunderstanding their service, but i fail to see the point. With put.io, a server can torrent the file for me. And then... i stream it from the server... So, my pc still needs that gigabits of transfer, at my usual net speed. Where was the gain? Couldn't i have used the same time to torrent it? In fact i would have to torrent it ONCE, but with put.io i have to stream (and thus effectively download that data) each time i want to view that file...
You also avoid nasty letters from copyright lawyers, which can get you kicked off campus at some universities or even kicked out of school sadly. Of course a vpn also avoids this, or a seedbox, or newsgroups
The biggest advantage from other commenters seems to be that put.io has a fairly large user base and if another user has already downloaded a torrent you want, it's available instantly in your put.io account.