Please visit your favourite torrent site and get a torrent or magnet link.
If you don't know what these are, here's a very simple primer: https://medium.com/@hayliminok/a-very-basic-primer-on-torrenting-b8708a143f6f
When you find the torrent or magnet link that you want, right-click on it and choose Copy Link Address.
Now in put.io, click the New Transfer button and select Paste Link. Paste the link you've copied inside the text area and click Start Transfers.
This is all you need to do to start your transfer. The progress of the transfer will be shown in your Transfers page. Once the transfer is completed you can click the Go to File button to see the contents of the transfer in your put.io storage.
Here is a short animation of all that in progress: