Good question! Saying your name or nickname followed by out means you're done or leaving that place/situation. It is similar to a mic drop which is "an emphatic and declarative gesture signifying the conclusion of a performance of note, often literally (or as if) dropping a microphone." - Webster. I believe this slang originated from walkie talkie language. When finishing a sentence over walkie talkies, you say over when you finish a sentence and your name and out when finishing the transmission. Basically he is saying I'm done with this conversation and bye. Hope that helps!
