"Song"? Is every piece of music a song?
I rarely listen or play anything other than 'art' music - namely classical (yeah I'm a snob) Regarding meaning, for a full discussion read "Emotion and Meaning in Music" by Meyer*. For me I don't search for 'meaning', whatever that means, in music. I play music because it puts me in a different place that is still connected to this one. I explore the musical textures and ideas, and feel them viscerally.
Last night I played a Chopin Prelude - the Db major. Slowly - langsam, each note reluctantly giving in to the next. Mostly I explore the tonal range that my grand is capable of producing, seeing how quietly and then thunderously I can perform (forte-piano, right?)
I also explore why the composer chose these notes. Either by accident or intent I look at different chords, or a clink in the melody, or just changing registers. Each change weakens the piece. This fascinates me, somehow this prelude seems to be build into the structure of music and thereby the physics of vibration and of the universe. By extension that applies to me too, since I can appreciate it.
* This is a book for outsiders, most musicians I know hate it because they understand that music is a temporal/experiential phenomenon, not an intellectual one which stands outside of the moment. This is why I do music - it gets me out of my fucking head.