Noone "gets away" with copyright. You can upload whatever you like to YouTube but if you upload copyrighted content you'll get a claim, there different types of claims though, some can block the video world wide and some can allow the copyright holder to monetise and make money fro your video, the copyright holder also has the power to REMOVE your video and give you a copyright strike.
Most of the time lyric videos distort the music a little by making it a little faster, a different pitch, etc and it can dodge youtubes copyright system, but overtime it has got a lot smarter. Unless you have the rights to use the copyrighted content it's not worth using unless you are using it in a "transformative" way, this could be news reporting etc, but you can only use small segments of a clip/audio.
Hope it helps a bit.