When you have to face the music, you're confronted with the consequences of something bad you've done. If your boss catches you lying about what time you got to work, you'll have to face the music.
Anyone who lies or avoids a responsibility for long enough eventually has to face the music. This might involve simply admitting you've done something wrong, or being punished or scolded by an authority figure. There are a few theories about where this phrase came from, including a military origin — at one time, military officers were "drummed out" in disgrace after committing offenses. Another possible source is the idea of stage actors facing the orchestra when they walked on stage.