For this to work the following things need to be true:
NOTE: For this to work, you need to be on a Mac. Also, it requires that when you set your password, you clicked the "Remember this password in my keychain" check box.
To retrieve my password, I did the following:
1. Opened Applications>Utilities>Keychain Access.app
2. In the search dialog, searched for "iPhone Backup"
3. Double clicked on the most recent entry for "iPhone Backup"
4. At the bottom of the dialogue box, click "Show password" and type in the administrative password for the computer.
The password is then revealed.