[edwhitesell]

All of the "good" WiFi systems make it really easy to have information changed on the splash pages based on the Access Point (therefore the physical location, and coverage area, of the Access Point).

As @davismwfl speculated, they probably pull the flight status information from a data feed internal to the airline, or one of the third-party companies that provide this data. Then displaying it on the page is easy.

It's possible there are multiple gates served by one Access Point, and they may be listing the next flight to everyone. However, given the low cost of Access Points and cabling infrastructure in today's airports (vs. when I did this type of stuff 13 years ago), there's one Access Point at each gate (in the counter, above the ceiling tiles, etc). As long as you're not closer to a different gate, you'll always get the correct "next" flight.

Same SSID, low radio power outputs, non-overlapping channels (where possible) and most users would never know.