Lyric discussion by lingeron 

Cover art for The Meeting Place lyrics by Last Shadow Puppets, The

I've been listening to the entire album and I really think that the songs are all related to one relationship and kind of chronologically displays the events as the two people gradually realize that they were not meant to be together. It climaxes in this song which is I think the breakup portion, and this ties in to the next song The time has come again which deals with the guy trying to move on and accept the decision he's made to leave her behind. Listen to the album as a whole and you'll understand what I mean

This explanation is awesome and you're the only person who has noted that the whole album relates to one thing. I've got carried away with my own comments (see below) but you've totally nailed it.