There are currently no events that you can hook into within an Appointment or Message object from the Mailbox API that can give you access to real-time changes. In fact, I believe this FindTime add-in is using features that currently aren't publicly available from either the Office Add-ins framework, the JavaScript API for Office or the Mailbox API.
Note though their task pane is activated via a ribbon button/add-in command. My guess is that FindTime is using the Outlook Notifications REST API to get updates as the meeting recipients are updated by the user. However, that API is currently in preview mode and I don't think you can use this yet for a production solution. Hope this helps!