Instead of hitting FLY use the menu at the top of the screen to select RECORD TRACK (or whatever it actually says, I forget :-) to start the mission. Then when you quit/die/whatever you'll find yourself prompted for a name for the saved .trk file
Also make sure that if you are hosting, after you name the file and click SAVE dont just hit Enter... make sure you back out of the game all the way and then restart to view the file.. We have had problems with saving and not backing out and when you play the track.. your jet is the only on in the track.. Backing out fully has remedied this problem