When you do Apple Pay at a brick-and-mortar store, the phone actually doesn’t do very much. The phone simply communicates a token which relates back to the card on file. It’s the merchant terminal that is connected to a server and sends off the transaction. This is why Apple Pay can be performed on a watch or phone with no network connectivity.
With a standalone NFC tag, it’s the sticker that is offline and inert and the phone does the heavy lifting of making the transaction and sending it to a server to be processed. I think what is happening here, essentially, is scanning the sticker with your device triggers an Apple Pay on-the-web transaction in the background. For the user, it’s convenient not to have needed to visit a website, download an app or otherwise login beforehand.
The same result could be achieved with a special-format QR code that encodes the payment data but that would add friction like needing to position the code in the camera frame — and would preclude the watch altogether.
What wasn’t clear is whether the NFC stickers are proprietary and will only work with Apple devices or if this is all based on some kind of industry standard. I assume it is the latter, though, as Jennifer Bailey showed the stickers bearing the generic contactless logo and not Apple Pay branding.