Karaoke is (amateur) singing along to a song, usually a version of the song with the main vocals removed. When singing karaoke the lyrics to the song are usually presented on a TV screen and highlighted as they are to be sung.
In subtitling however, karaoke usually refers to those lyrics themselves. The karaoke effect refers to the way they are highlighted.
So, a karaoke effect is a visual effect synchronised with a song, used to help singing karaoke.
Reality has it that karaoke effects in the fan subtitling community today are used more for “eye candy” and showoff than the real purpose of karaoke subtitles, assisting the viewer sing along to the song.