Music Publishing is the business of protecting and promoting song copyrights and collecting the royalties that these copyrights generate. Think of a song as intellectual property (it's an original piece of art!), and music publishing as the business that makes sure that songwriters get paid a fair compensation when their intellectual property is used by companies. Those companies could be record labels (mechanical royalties), radio stations, bars, and restaurants (performance royalties), or film studios and advertising agencies (sync license fees), and many more. Note that music publishing pays royalties only to the writer of a song, not to performers (so Bob Dylan, not Jimi Hendrix, gets the music publishing royalties for, "All Along The Watchtowers" ™).