To enter passwords, first you must beat the game. Then, at the first Ranger Net screen, press L + R + A + X + Y + Left, and an option to enter passwords will appear under the "play a special mission" option.
Manaphy egg special mission:
Enter the code P8M2--9D6F--42H7 as a password. The mission is called "Recover the precious egg!".
Thanks to Revolution Readers Brian and IronLung!