In the video, the trigger is pressed down slightly before the final push releases it from the console. As the Joy-Con moves away, you can also see a small cylinder retracting into the top portion of the inside of the controller; To me, this indicates that when you press the trigger, you are actually pushing that cylinder out to help release the Joy-Con.
Here's a GIF of the relevant part of the video:
GIF by Andrew Liszewski/The Verge
The mechanism for removing the Joy-Con controllers looks similar to one that was included in a 3D-printed mockup by accessories maker Genki. brought to ces,