Each Swatch Bellamy watch has an NFC antenna and a chip integrated in its case.

The chip contains payment data similar to those which are integrated in payment cards (prepaid/debit cards). The NFC antenna allows these data to be transmitted to the payment terminal. 

Basically the payment functionality works exactly like it does for other contactless payment cards. When the payment terminal is activated and the watch is moved to a distance of less than 4 centimeters, payment data are transferred to the terminal and payment can be executed.