The Media Library section of iExplorer is read-only, so you won't be able to delete any music on your device in that section. iExplorer does let you to delete music where the actual audio files are stored in the Media folder but deleting those files in iExplorer won't update the music database, so there would still be references to the deleted tracks in the Music app where they would be grayed out and unplayable. This is due to the way that iOS works and how Apple designed it, so you'll need to use iTunes for removing and adding media such as music. That way, both the music files and the database references will be removed. If you don't like syncing with iTunes, you might consider enabling the option to "Manually manage music and videos" which will allow you to simply drag and drop any tracks to your device or remove them without syncing:
Or, if you are using an iPhone, iPad, or iPod Touch, iOS 7 and 8 added the ability to delete tracks directly on your device in the Music app. iOS 7 only lets you delete individual tracks, but iOS 8 lets you delete entire albums or entire artists too. Just swipe left on any track, album, or artist and tap on Delete.