Wordorigins.org

This phrase meaning to accept responsibility, suffer consequences is an Americanism dating to the mid-19th century. The underlying metaphor is uncertain. From The National Era, 13 July 1848:

Why, stand up and face the music.

There are a couple of common explanations for the phrase. Both are plausible, but which, if either, is the correct one is not known. The first explanation is that it derives from the stage. With the musicians in a pit before the stage, to face the music is to turn towards the audience and either their hoots or cheers.

The second is that it is military in origin, and refers to a ceremony where an officer is cashiered and is literally drummed out of the service.

(Source: ADS-L)