To restore your data from a backup you made from Contacts Journal, you should first ensure you are not syncing Contacts Journal with iCloud. Then:
- iOS: go to the More -> Manage Data -> Restore over WiFi page. You need to have another computer connected to the same Wi-Fi network as your iPhone or iPad. The URL to open on your computer will be printed on the Contacts Journal screen. Upload the backup zip file on that website
- Mac: from Contacts Journal preferences -> Manage Data -> select "Restore", and select the backup zip file that you wish to restore.
If the backup file is password-protected, you will need to enter the PIN to complete the Restore.