It's a crappy video, not sure why they chose that specific one. Here is a better one: https://www.youtube.com/watch?v=n_QyPCCxriQ
TLDW: Use your thumb and forefinger in a flicking motion to add a large amount of spin to the cork so that it rotates very fast with the feathers pointed up during rotation, like a top. While it is spinning like a top in midair, make contact with the racquet, but not too hard. The initial contact to send the shuttlecock over the net turns it upside down (cork points to the sky).
The rotational motion that you added (with the flick) to the shuttlecock adds gyroscopic stability, like the rifling on a gun, which keeps the shuttlecock upside-down for longer. When the shuttlecock gets over the net, the cork is still pointing to the sky which makes it impossible to return because the racquet can now only make contact with the cork in a downward position.
In normal flight/serve, the cork is pointing to the floor when the shuttlecock gets over the net, so it can be given some lift when returning.
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[1] It also doesn't help that the camera view of badminton matches is not from the side, so you can't really see how hard this is to return even in slow motion.