오다 here is a message arriving, and the 라고 clause is what it says: DM이 뭐라고 왔는지가 중요한 거죠 (what matters is what the DM said), 아빠가 '알겠어' 라고 오더라고요. The same shape covers a person turning up with a stated reason — 쯔양 님 팬이라고 와서 (he came saying he was a fan), 같이 놀아달라고 올 때 (when he comes asking to play).
Register: casual; the messaging use is very common