Rent Gates

You will need to rent gates at airports before trying to create a route to that airport. Otherwise, the desired airport will not show in the dropdown menu when trying to create a route.

Navigate to the Rent Gate area by going to Gates, and click Rent Gate. You will now see a dropdown menu with which you can select a way to view the gates for rent.

By the way, the map you see in the bottom of the screen shows all the airports that are currently in-game. Apart from showing that, it doesn't do anything.