Hi A K,
Thanks for getting in touch, interesting request here ๐
To sync with other calendars (or sites) you can use the ics (ical) feed from the original site where The Events Calendar is installed (http://sitename.com/events/?ical=1). To show this on the external site (WordPress or not) you should create a parse script to display the feed as a list (PHP ics parsers).
Does this makes sense? If the other site is in WordPress as well, you can install The Events Calendar + iCal Importer add-on, which can run scheduled imports of the original events feed.
Please let me know if you have any other questions on this,
Best,
Nico