As for local first, I see it as a combination of a few things
- moving databases as close to users as possible [0]
- writing data without coordination with other nodes
- background sync
What this buys you is extremely high availability (as you can write without having to coordinate over a network), lower latency in general, because the database is right there (ideally within visual range), and more reliability, because network outages only effect background sync.
I'll also add that while local first is often seen in the context of b2c users, i.e someone using their smart phone while travelling on an urban train system that has network dead spots - I think the real use-case is industrial. A lot of facilities completely grind to a halt when there's a network outage, which is not that infrequent. To say nothing of in field and IoT applications in industries like agriculture and mining.
[0] the logical conclusion of "edge computing"