How does 반갑다 become 방가방가?
I know the greeting 방가방가 means “nice to see you,” and that it’s quite casual. I learned that it comes from the verb 반갑다, but how does this work? The 니은 in 반갑다 becomes 이응 in 방가. Why does this happen?

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Hi Juno,

It's an online chat slang like at least 10 years ago. Nowadays, we use it for making fun.
There is no such grammar. It just sounds cute.

Thanks

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It is called '비음화'.
(It is slang so it is not grammatically right but it follows grammar.)
'비음화' means to become(=화) a nasal(=비음).
ㅁ, ㄴ, ㅇ are nasal in Korean.
ㄴ is already a nasal so it is technically not 비음화 but it makes sense.
ㄴis changed to ㅇ because ㅇand ㄱ sound by the same vocal organs.(they are called velar/soft palatal consonant.)
So it is for the convenience of pronunciation.

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Agreed with the other comment. 
I think you should really only use this if you're trying to make people laugh in a very casual setting haha.