One of the comments I have received about The Portable Galactic Empire from players is that it lacks a story. That is, it lacks a story in the box. And although this may change in the future, for now I'm OK with that.
As a player, I personally prefer games which allow me more latitude to create my own narrative. In fact the "openness" of the game directly relates to how much I enjoy a game. In this case I have built a game through which several players may construct their own stories.
If you want to see a more storied session of PGE, try doing this. Before play begins select a theme for your play style. e.g. "I've decided that my race is made of entirely artificial entities which share a mind link." Now start considering how that is reflected in the mechanics.
The spending of influence, in this case, represents the alignment of processing nodes. In order to launch an invasion, a sufficient number of processors must agree to the decision. Monuments are a manifestation of the artificial being. 5 monuments represent the true form of the data. And so on.
Try it yourself. I'm sure most of you are capable of coming up with much more interesting stories than I could have possible put in the box.