Diagnosis
The router itself is functioning and broadcasting WiFi, but its management connection back to Google's servers (and thus the Home app) is broken. This is usually a cloud sync issue, not a local network problem.
How to Fix It
The fix is to force a local reconnection. 1) In the Google Home app, tap on your router device. 2) Tap the settings gear in the top right. 3) Tap 'Remove device' and confirm. Don't worry, this doesn't factory reset your network. 4) Now, with the router still plugged in, tap the '+' in the app to add a new device. It should find your existing network. Follow the prompts to re-add it. This re-establishes the app's control link.
If That Didn't Work
Your router's firmware might be stuck. Perform a hard reboot: unplug it, wait 30 seconds, then plug it back in and give it 5 full minutes to restart and try to reconnect before opening the app again.
Heads Up
This 'offline' error in the app can sometimes happen after your ISP changes your public IP address. The re-add process in the fix above is the standard solution and won't mess with your WiFi name, password, or any other settings you've configured.