Collect the special item crate in the indicated level, then complete that level to unlock the corresponding special item:

Alien Hat: Level 2-1
Astronaut Hat: Level 1-1
Babe Speech-Bank: Level 4-3
Bolt Flag: Level 6-4
Captain Speech-Bank: Level 2-3
Column Gravestone: Level 6-5
Creepy Speech-Bank: Level 1-3
Cross Gravestone: Level 6-2
Crush Flag: Level 4-4
Digital Gravestone: Level 4-2
Doctor Speech-Bank: Level 3-3
Eyeball Hat: Level 1-5
Female Computer Speech-Bank: Level 5-3
Helmet Hat: Level 3-1
Insignia Flag: Level 1-4
Logical Speech-Bank: Level 2-5
Meteor Flag: Level 3-4
Obelisk Gravestone: Level 3-2
Padded Hat: Level 4-1
Petrified Gravestone: Level 4-5
Post Gravestone: Level 2-2
Princess Speech-Bank: Level 6-3
R.I.P. Gravestone: Level 1-2
Red Hat: Level 5-1
Slab Gravestone: Level 5-2
Strong Worm Flag: Level 2-4
Tentacle Flag: Level 5-4
Visor Hat: Level 5-6
Yellow Hat: Level 6-1
Zap Flag: Level 3-5