You need to upload the files to some place. He's saying with app net it's possible to upload it to the network (like Twitter) itself, and not to some third party storage like dropbox/mega/box. Just like sending images on Twitter.

And regarding "just send the link via DM", that's currently not possible any more: https://dev.twitter.com/discussions/23044

You have to make a dropbox account to see my file, if the file isn't public.

Edit: wait, maybe that's not true. It's even worse - anyone with the shared dropbox link will be able to get at my file even if they're not authenticated as someone I want to share with.

How is not requiring registration "worse"? Just don't send the link publicly.

Sure, the recipient might share it, but s/he could also just download the file and share that, so you haven't lost any security.