Hi there and welcome to the Community, @user29448 
Our Appointment Booking widget requires integration with Google Calendar to automatically mark booked slots as unavailable:
Here’s a tutorial on how to connect your calendar: Syncing Google Calendar with the Appointment Booking Widget
Please try it out and let me know if it worked.
As for the option that lats users cancel booking, this idea is already on the Wishlist. Feel free to upvote it 
- End user can change or cancel a booking