Means the same as, "I
don't give a flying
fuck", "I
don't give a Funyun", "I don't give a rat's patootie", etc.
This is a clean way of saying it -- though that hankie really needs to go in the wash. :-)
WarGrowlmon: What? Beelzemon?
Taomon: What are you doing here?
Rapidmon: Looking to hit us while we’re down, huh? That’s your style
right?
Beelzemon: Hey, trash me if you want but I swear I’m here to
help.
Rapidmon: Do you even know the meaning of the word?
Beelzemon: I’m on the level fellas, really.
WarGrowlmon growls at him.
Beelzemon: But I don't give a dirty handkerchief if you believe me or not, cause my partners believe in me.
Flashback to Mako giving Impmon his
gun.
Mako: You can
blast all the bad guys with it.