If you don't know which file to download, you want BackgroundMusic-0.4.0.pkg and you can ignore the others.

Changes since v0.3.2:

• Make sure the installer launches BGMApp as the logged-in user. (4c7eba30afb7c19567e1920dc20d9f5558b7da37)
• Add app volume workaround for Google Chrome. See #61. (dfad77dc35cd065390cff0d870ed6bcb81042d7e)
• BGMDriver: Copy settings when a new client matches an existing one. (f61f998c5142f05fc26f412327d28bd4d7e8aece)
• Fix stale data when recording system audio #217 (40f0128dcd5319da6f1a1795c1b733a26c6fe675)
• Add optional debug logging in release builds. (2c1677305d5ce9b932965be4e4cad4187ab83610)
• Hopefully fix a segfault caused by BGM_PlayThrough. (488ed16dfce7950ac6d0aef2cd904abbe12e097e)
• Fix apps failing to start audio after upgrading Background Music. (70a64de73428820e345dbfb5b33ec51b5267afa5)
• Try to avoid using the discrete GPU on multi-GPU systems to save power. (f24a6cb2f12b35820dfc825d58d7340b72f618ba)
• Workaround deadlocks when starting IO on macOS Big Sur. (c0ab98b98b0f2fab1f1fb4ddf1b19c807e71c972)
• Added app volume and pan setting to AppleScript #457 (9765193c1fb42d28bbfc1d881424daf554e999a6)
• Add a bundle ID property for apps in the AppleScript API. (213c339309b4f65206e82a388375ed73ddc5af13)
• Installer: Retry if opening BGMApp fails. (0e0cc5fd05cf14454137ac09424897a2054546f5)
• Fix for issue #475 dropdown arrows move instead of doing something (24b248e09d727db5fdf1ca1a33efc839d537f1e7)
• Fix installer errors on macOS Monterey due to removal of Python 2 (01217d2d8c4a2a7e4fbe0c7d8e2864aa91a13fb1)
• Add app volume workaround for Microsoft Edge #597 (4608f4bbac3c1e6a1609efd596e3f021fae6bb9d)
• Installer: Fix post-install check for BGMDriver failing unnecessarily. (b2770f1793680150fa6138ed75c3a9bc5406d997)
• Show an error message if the user denies input device permissions. (89eebb0e021a8b438869ab9e563f19565e189299)
• Fix slider knob sizes. (d0482871409f1f90d16e0ec5b8e859f3ac6e03dc)
• Drop support for macOS 10.9. (d0482871409f1f90d16e0ec5b8e859f3ac6e03dc)