From a technical point-of-view, how is the Apple Watch able to unlock doors?
To put it simply, the Apple watch acts like a key fob.

You've probably seen these credit card "keys" or these fobs on a key ring that people put up to a sensor on or next to a door to grant them access.  (HID is the dominant market player - I just grabbed a couple of images from their site).




These devices just a have a very simple RFID transmitter in them that sends out a code, usually Wiegand, when put close enough to a reader.  The reader communicates back to a server with the code that was presented and send either a "yes" or "no" back to the reader which then operates a door lock.  This is basic access control.. Below is a (very) simple block diagram of how a typical system is laid out.  

The controller (blue box) handles the reading of the cards from the sensor (gold circles) and the (un)locking of the doors.  In fact, the controller has a large bit of memory stored so that permissions can be pushed to each door and the system will continue to function even if the server is down.



The Apple watch (and iPhone) has the ability to "read and write tags" with their NFC chip.  So, the NFC chip emulates a card or keyfob by transmitting the ID code.  The door reader still communicates back to a server to get authorization to grant/deny access.  So, just like the key fobs/cards don't need any special software or apps to function, neither does the Apple iPhone/Watch.