It looks like they basically ask whether you want to use the site or the app the first time you visit on a device, and if you choose the app, it sends you to the App Store and sets a localstorage setting that triggers a redirect on subsequent visits. (Edit: OK, it's a little bit more sophisticated than that. But I think that's not too far off.)
(I'm not sure how they're detecting whether that hook is registered or not, though.)