What does 동떨어진 소리 mean?

ANSWER: an absurd remark

동떨어지다 means to be far apart or to have a space between and can refer to the separation of places, things, or relationships.  By itself, 떨어지다 means to fall or to drop, but it can also mean to separate or to be detached. As for 동, it seems to come from the Chinese character 同 (동), which can mean either the same, a group, or together. That would mean 동떨어지다 would literally mean either to be separated from someone or something that is the same or to be separated from a place or a group. Here are some example sentences using 동떨어지다 from my Donga Prime Korean-English Dictionary:

• 5마일이나 동떨어진 곳
  the place 5 miles away from here
• 두 사람의 나이는 너무나 동떨어진다
  There is such a great disparity of age between the two.
• 그 길로 갔더니 가야 할 길과는 동떨어진 데로 나오고 말았다.
  The road took us far out of our way.
• 그건 이것과는 동떨진 추측이다.
  Your guess is way off the mark.

Considering the 동떨어지다 meaning, "동떨어진 소리" could literally translate as "noise separated far from [the conversation]." Here are two example sentences from my book of Korean idioms entitled "뜻도 모르고 자주 쓰는 우리말 숙어 1000 가지":

• 그 친구는 가끔 이야기 중에 동떨어진 소리를 해서 듣는 사람들을 김빠지게 한다.
  That friend sometimes makes absurd comments during a conversation, which causes other people in the conversation to lose interest.
• 그는 나이로는 아버지뻘 되는 조카에게 어떻게 말문을 열어야 할지 몰라 동떨진 소리를 했다.
  Because his nephew is about the same age as his father, he was not sure how to start the conversation and said some absurd things.

Be careful to say 동떨어진 소리 and not 똥 떨어진 소리 because the second phrase means the sound of falling poop.