You can tap on either an Airport marker in a displayed route or on a map data marker (e.g., Flight Rules, Temperature, etc.). Doing so will bring up the airport information pop-over. Tapping on the FULLSCREEN button will take you to the Airports page where you can view the information in fullscreen view.
You can also tap and hold on an airport shown on the map. From the pop-over, select the AIRPORTS button on the bottom and then next to the airport you want to view information for, tap on MORE and then DETAILS.