In order for you to receive a notification of a booked appointment, you need to log into Calendly and sync your Calendly account with the calendar that you use daily, such as google calendar.
The process is fairly easy. When you initially log in to Calendly, it will pop up an option for you to sync your calendar and you can simply follow the steps provided.
Also, you can scroll to the top right corner of the home page after logging in and click on Account and scroll down to calendar connections. From there you will be able to add a calendar account.
Once this is set up, you will be notified of any new booked appointments.