My father is supposed to drop me off at the airport on his way.
Literal
Father [subject-が] incidentally me [object-を] airport [until-まで] send-for-me thing [result-に] has-become [polite-いる].
ついでに means 'while at it / incidentally / on the way.' ~てくれることになっている combines benefactive + established arrangement: 'it's been arranged that father will (kindly) take me.' まで marks the destination endpoint.