The karaoke company negotiates a fee for the master use license of their recorded 'arrangement'. If visuals are seen (lyrics), those may be negotiated with the karaoke also as the visual of the lyrics is their property too (not the actual lyrics themselves).
Rates depend on the market (i.e. local, regional, national, worldwide). In my experience, I've had master use licenses for small markets at $100. Worldwide in perpetuity for a movie, tv show (like American Idol, AGT etc) for $3000-$4000. It's a nice payday for me, but it takes forever to get the actual check
.
And that's on top of what the writers/publishers who actually own the song itself will get. My fees are only for my arrangement of the 'popular' song that was recorded for karaoke.