It means, literally, “upon flagrant transgression” - though flagrant deliciousness sounds much more positive…!
A simple glance at Wikipedia would inform you that it refers to a “blazing crime,” i.e. one that is obvious at the moment.
What, did somebody just whack him with it?
Now I’m wondering where the word ‘delicious’ came from…
In his short story “A Separate War,” Joe Haldeman tells of a naked and, um, excited tribesman who chases a woman, apparently intent on raping her, but she kills him with a sword. He ejaculates as he dies, never having touched her.
“Delicious” comes ultimately from the Latin delicere, to allure, charm, entice, delight.
In flagrante delicto, by contrast, involves the past participle of the verb delinquere, to fail, to be at fault.