Collect thy transfer by transferring thy transfer to Collect. Fortunately, the process of saving a transfer to a board isn't as difficult or as weird as that sentence.
If you click on a transfer link on your mobile device, you'll get prompted to the download screen which suggests saving the content of the transfer directly into a Collect board.
After saving the file, you can move the items to an existing board or make a new one.
There is a guide on how to store them directly to your device in this article:
Currently it's not possible to save a transfer directly to Collect if you're using Collect web.