When I learned the word busybody the first time, I was in 5th grade. It appeared in a story I had to learn for class. I figured it meant someone who was very busy, and didn't bother to look it up. When the teacher asked us if I had looked it up as I was supposed to as part of my homework, I lied and said I did. When I told her I found the meaning "in a dictionary" she asked me to bring her the dictionary. Needless to say, I was a very embarrassed 5th grader caught in a bad lie.
Etymonline has this:
busy O.E. bisig "careful, anxious, busy, occupied," [...]. In M.E., sometimes with a sense of "prying, meddlesome," preserved in busybody.
So, the word busybody didn't "end up meaning" what it does; much rather, it had that meaning all along. It's busy that used to have an additional meaning it no longer has in contemporary English. (If you read that Etymonline entry, you'll notice that that's not the only meaning that busy used to have but no longer does except in the idiom get busy: "The word was a euphemism for 'sexually active' in 17c.".)
I think it has something to do with the origins of the idiom. It did not originate as a word, but as an idiom. It must have been a euphemistic way to refer to busybodies, and people in the same social circle must be using the word to describe their friend in a public setting.
Of course, all this is my guess. Please correct me if I am wrong.