In order to use your Apple Watch to unlock 1Password, you'll need to adjust some security settings on your Mac. Click Set Up Apple Watch and it will automatically open the Security & Privacy settings in System Preferences.
Now, if it's not already checked, click the box next to Use your Apple Watch to unlock apps and your Mac. You'll have to enter your password, but once this is complete your computer will be ready to be unlocked using your Apple Watch.
To use this on an ongoing basis, you'll have to ensure that both Wi-Fi and Bluetooth are enabled.
Next time 1Password is locked, you can press the Watch icon (or Touch ID icon) to prompt you to unlock from your Apple Watch.
On your Apple Watch, simply double click the side button (underneath the dial) to unlock 1Password.
1Password has two browser extensions: 1Password and 1Password X. The former requires the macOS native 1Password app to be installed while the latter requires you to be authenticated in the browser.
If you're using the normal 1Password extension, you will be able to unlock using your Apple Watch, but if you're using the 1Password X extension for something like a Raspberry Pi, you won't, since it's a fully in-browser app.