What system populates the Flight Stats?
The same system which populates all the data on displays in a Railway Terminal.
For a standalone airport, this data can be mined from ATC of that airport (Which has all the data regarding incoming/outbound flights ).
On an International scale, the data has to be mined from various different agencies. According to a data provider FlightStats:
It gets data from:

Complete, accurate data is the cornerstone of our business and what
sets us apart from competitive solutions:
Geographic Coverage - FlightStats provides definitive information for approximately 99.5% of U.S. flights, and better than 86% of
flights worldwide.
Completeness - FlightStats queries multiple sources to create a record for each flight enabling us to offer a broader range of
information (for example, gate information).
Accuracy - We have invested heavily in the areas of parsing, interpretation and error checking and developed the logic that enables
handling of difficult issues such as cancellations, diversions and
changing schedules.
Codeshare Support - Our codeshare logic enables us to deliver flight information for both the operating and the marketing carriers,
filling what is often a major gap in coverage.
Real-time data sources include:
FAA ASDI Data Feed
European Data Feed
GDS (Sabre, Amadeus, Apollo, Galileo)
Direct Airport / Airline Data Feeds

Batch data sources include:
Innovata Schedules
TSA Security Wait Times
Security Information
Health Information
Consular Information


For the system which gets all this data from various different sources, its the central computer database of the airport, which gets data from ATC and the a fore mentioned sources, and this database is accessed by many other end users, including the Screens in the airport