No, there is no monthly fee! It's your Wi-Fi and most of us are already paying quite a bit for that! At this time, cloud storage storage for a rolling 7 days of history is also free of charge and is included. Once you sync your SkyBell you will see "History" in the app. You can save history files and screen capture pictures to your mobile device for longer term storage.
Related article: How does the Activity History work?