How does the Airbus auto takeoff system work?
This is a quote from Airbus's press release:


  Rather than relying on an Instrument Landing System (ILS), the existing ground equipment technology currently used by in-service passenger aircraft in airports around the world where the technology is present, this automatic take-off was enabled by image recognition technology installed directly on the aircraft.


So it sounds like the plane is using the same technology used in autonomous cars, which are able to use computer vision and cameras to navigate traffic.  Except in this case the plane just had to keep center-line and rotate at the appropriate speed.

It is also shown in Airbus's video, ATTOL: Autonomous Taxiing, Take-Off and Landing test flight: