The Live Activity appears on your Lock Screen and Dynamic Island while a focus session is running, with the time remaining. Tap it to jump back into Aura. It's the reason you can lock your phone during a session rather than watching a timer.
The Stats widget puts your week's screen time on your Home Screen. Long press an empty area, tap the plus, search Aura, and pick a size.
If a widget looks stale, it's iOS deciding when to refresh rather than Aura failing. Opening the app updates it.