In 0.8.1, files are shown in the music menu with their filename.
Drop the music files into that directory.
Create new directory custom_bgm/ in your DDLC/ directory if it does not already exist.
Ensure that the music you want to add is either MP3, OGG/VORBIS, or OPUS format.
Version 0.8.2 added support for additional music formats (MP3, OPUS) as well as looping via metadata tags ( #1670, #1695). Version 0.8.1 added support for custom music (OGG/VORBIS) in MAS ( #1592).