This phrase is used in an unapologetic way when describing events that range from the embarrassing to the indescribable stupid, but for which the person involved feels no remorse because it presumably brought the subject enjoyment or amusement. Usually accompanied by a ridiculous .gif to convey how much the person does not give a fuck what others think.
Phrase originating in it's current form, like everything, from a Simpsons episode.
friend 1: "dude, what the fuck. did you really go home with that ugly girl last night? she had no teeth and was wearing a wig, sideways."
friend 2: "I regret nothing"
girl: "I stayed home last night. I cuddled with my cats, drank 4 beers and ate six tacos. at 2 am I ordered pizza and fell asleep with my hands inside my pants in the living room. I regret nothing"