Music and Harmony

So I was listening to a new song on the radio.  It's called "Cough Syrup" by Young the Giant.  The first two lines establish the melody of the song.  I'm someone who is very used to harmonizing with music, frequently singing along with the radio, in harmony notes, rather than the melody.  So, that's what I was doing in my head, and out loud.  Imagine my surprise when the 3rd and 4th line of the song was the harmony notes instead of the melody notes.  It kind of surprised me a bit.  I like the song, find it catchy.  It's now on my iPhone.  Still, I feel like I'm not sure how to harmonize along with the song.  Do I go back and sing the notes from line 1 and 2 when I'm with lines 3 and 4?  And vice versa?  I end up getting so wrapped up in the notes that I completely lose the words.  So I honestly have no idea what the song is about.

And, of course, the music nerd in me appreciates this picture.  Looking at it first, I was trying to figure out what was up with it, aside from the changed key signature and time signature.  I thought i needed to note that the time signature was off on the red notes.  Then I realized I just needed to read the tune of the red notes.  I almost printed it out and sat at my piano, until I thought about it.  Then I felt gratified that I know how to sight read (vocally) music.  Not that I can sing that high at all, which is why I resisted at first.  I still should probably sit down and figure out what the tune is in the black, but I'm lazy.