Can someone explain how and why I would truly want to link out to another app on someone's phone? I get the whole android intents thing and having a "share" button and letting other services handle some of those things, but in what cases is it super beneficial to link to another app and leave your app?
The only thing I can think of are things like PDF viewers or maybe cross linking your own apps?
Sure. So let's say you run Skype. Currently the way skype works is to add somebody, they have to find out their Skype name, tell it to you, you receive it, they type it into a search box, pick from a list, then wait for the other person to confirm. That is, shall we say, terrible.
Deep app links would let you just send somebody skype://user/jim89 or better yet skype://invites/9384jr83 which provides a massively better on boarding experience.
Not only that, but I am working on an app where you don't even have a username, so there's nothing to type in a search box. A URL-based scheme like this one allows you to eliminate an entire field and the whole "create an account" step.
So one of the examples they use is really powerful: buying movie or concert tickets.
I'm in an app like IMDb or Facebook or whatever and I see a movie trailer. I'd love to be able to buy tickets immediately. Right now, unless the app developer specifically hard-linked to a ticketing app, I have to manually open that app, search for the ticket, then return to where I was before. With an app link, I can simply open that movie page inside MovieTicket.com or Fandango or whatever, and buy my tickets (using whatever logins are already associated with that app and account) and then pop back to where I was before.
Same thing if I have a special type of audio or video file in a text document. Rather than having a link to that file open in a web browser that then tries to open up inside an app, it can open up inside the native app, and then pop back to my document after it is done playing.
It's not that different from the "open in" option in iOS or intents in Android -- except you can also have that sort of stuff apply to web markup. So a link to a content type accessed on the mobile web will still execute in-app, rather than in a mobile web view, which might not be as full-featured as the content owner wants.
I have a photo collage app that produces a jpg. The user usually wants to share the collage on social media like Facebook or Instagram. Keeping up to date with the various APIs is a pain in the ass (and I use ShareKit, a library to aggregate these services), and the user still has to sign in to give credentials to import photos or post to social media.
Linking could simplify the exchange of data here, where I could just pass the photo to the relevant app to handle.
The people who want it most (will pay the most and will pay immediately) are advertisers. Which means the apps that want it most are those that have loyal users, but have been unable to effectively monetize them to advertisers. There are plenty of applications for it outside of ads. To extend your question - why would you truly want to link out to another webpage in someone's browser? That's what AppLinks/OmniLinks/DeepLinks are designed to do - to make the relationship between app use resemble the relationship between web pages.
The only thing I can think of are things like PDF viewers or maybe cross linking your own apps?