This is odd...

I get the exact same thing!

I listen to A LOT of incredibly complex/alternative music. A lot of which have multiple meanings and analogy to the lyrics. But even though I 'get' the songs, I find that while subconsciously paying attention to the song (driving, and yes, reiterating songs while trying to sleep) the meaning comes through even more crystal clear/instantly.
This is because your subconscious mind takes in a vast amount of information. Even more surrounding than when you're paying very close attention. So something like trying to fall asleep or being somewhat in a trance like state (moments before sleep, driving, showering, doing any cathartic task while listening) your mind produces information it has already attained, much more clearly. Most of the time while paying close attention listening to music, you're trying (or trying without realizing) to take in almost every word at every moment you hear it on top of the fact that when listening to a song, there's a multitude of sounds you're taking in, as well. Causing your mind a small sense of feeling overwhelmed. So in a situation like sleeping, or drifting off/driving, a specific lyric and sentence will hit you. But mostly because it's just formation of any specific sentences and them 'by itself' -- without the multitude of distraction music has that coincides with hearing lyrics (the beats, bass, guitar, drums) and every other sound that your mind is taking in WHILE lyrics in specific tones are being 'sung') ...

What I said in a nutshell; There's so much element that makes a song, more than just lyrics, so when you're actually listening your mind is distracted by so much all at once and when you think of JUST a sentence, it becomes much clearer to take in and understand.

Try reading just the lyrics to most songs, without singing the song in your head. You'll find you get most/all songs, it's more like reading a poem. Your subconscious stores a lot of information as well. Making the moments before sleep a very capable tool of understanding things.

write it down