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

Answer

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.

Answer

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.

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