Other phones with Briar installed can carry your (encrypted) messages, as in a game of whisper. This works best if enough people between you and the recipient had Briar installed ... but most people don't.
But I see how this feature could be very helpful if a state shuts down internet connectivity or during war or a natural catastrophe. The nifty thing is that the app can be shared from one device to another, so you are not dependent on having the app in advance of an emergency.
Ideally, everyone should have this installed as an insurance :)