Exactly what the trophy says - You need to shoot down 300 planes while playing in Campaign mode.
There's no counter in the game, but you'll most likely get this naturally before the game's over. Fighters shot down in the DLC campaign levels also count for your total, so there's no need to replay levels if you didn't get it yet by the end of Act II. It popped for me towards the end of Eastern Sands on the X360. On the PS3 it popped after I was done with the main campaing and DLCs, when I was redoing a few levels to get more ranks and badges. I'd only done five or six levels over, so if you didn't get this by the end of the main game and all the DLC missions then you should at least be close.
The trophy will unlock at the moment you shoot down fighter number 300, you'll get it immediately and not just after finishing the level.