I think I stole the idea from a Star Trek episode from The Original Series, "For the World is Hollow and I have touched the sky" (http://en.wikipedia.org/wiki/For_the_World_Is_Hollow_and_I_Have_Touched_the_Sky), and extrapolated how I thought that situation might evolve.
I see generational asteroid "ships" like this happening more by chance than as some Grand Plan to inhabit the galaxy. Asteroid mining seems like a plausible economic endeavor for our future, and after asteroid miners go through longer and longer shift rotations the asteroids may evolve into orbiting communities. After a long enough period of time they may find themselves to be self sufficient, and grow tired of listening to Republicans and Democrats blame each other and decide to "fire up the Ion Drives" and go for a galactic Walkabout.
Conversely, such "ships" from other (possible) alien cultures may be out there already, slowly plowing their way through empty space, seeking neither contact or attention and going nowhere in particular.