This term was originally used to name an event in which a group of people all rock out to the same song playing on headphones. However, onlookers cannot hear the song in the headphones and therefore, the group looks ridiculous. While participating in a silent disco, if you turn off/take off your headphones, you can hear everyone messing up the words to that particular song. Therefore, this phrase NOW refers to someone completely messing up the words to a song, but are convinced that they know them, and therefore, persevere.
Joan: "R-E-S-P-E-C-T camelbacks in PNC."
Rick: "Shit, you totally just Silent Disco-ed that song."