It’s actually quite a complex function. The announcements are made through the site wide combined public address and voice alarm PAVA system using the system speakers dotted around.
The clever bit is the linkage to the Airport Operations and using that data to provide automated multiple local gate announcements.
These announcements can be restricted in their bleed across large open areas as well as other gates, so that people who don’t need to know are not alerted.
Keeps them happy eating/shopping and the tills ringing.