Source data usually from the same system that does the airport's own internal flight information system, and it also feeds things like BBC Ceefax (though what gets displayed is differently formatted on each). As far as am aware there are no standard software packages for this, and each airport's IT team tends to do their own thing.
Sometimes things get through that show the logic is not perfect. Freight flights, GA, etc showing up sometimes.
Some airports (particularly Stansted) are bad at loading up their schedule too far in advance and then showing flights not operating on the day, and never planned to operate, as "cancelled".