There are 2 kinds of loading screens: the one that requires you to press any key (custom), and the one that starts when all players have finished loading (melee).
If you still want them to press any key, but just don't want to show the message: Hell_Master is correct.
(Note: only change the message that changes the "Press Any Key", I don't recommend changing the other value).
If you want to start the game as soon as everyone's done loading, then you will have to change the loading screen to the melee one ("Use Default Screen").
This will show the player loading progress and you will not be able to put any text in the loading screen, so that might not be what you want.
You can still customize the background, though.