Hello!

There are 2 modes:
- Online cloud sync: paste sync code from an Authi.me desktop or mobile application and your TOTP codes would be synchronized across devices. That is what "Paste from clipboard" button do.
- Offline: import a backup file that stores the codes (there will be no sync and the extension would work completely offline). Click "Import" if you want to use that one.

Backup files are interoperable: you can create the file in Authi.me on your computer or phone or use another application that supports export.
The file should look like:
otpauth://totp/username:?secret=t0rwx1csxkw73kryyogn8o6qykimclr2&issuer=service1
otpauth://totp/username:?secret=o6qykimclr2m6rwx1csxkw73kryyogn8&issuer=service2

Popular apps that support exporting offline backups are: Aegis, Ente, 2FAS, Bitwarden.
If you want to use the extension in offline mode: try Authi.me app or any of those apps.