To exaggerate or embellish a story or rumor by adding extra details, making it seem bigger or more interesting than it is.
彼はその話に尾ひれをつけて、実際よりもっと大事件のように話した。
He embellished the story and made it sound like a much bigger incident than it actually was.
噂は尾ひれがついてどんどん大きくなるものだ。
Rumors tend to grow bigger as details get added.
友達の失敗談に尾ひれをつけて笑いを取ったが、本人はあまり喜んでいなかった。
He embellished his friend's mishap to get laughs, but the friend wasn't very pleased.