What happens to pasta dough when it is run multiple times through the machine?
You're alluding to the correct answer:  running it through several times basically kneads the dough and develops gluten. 

Gluten provides strength to dough.  It forms a flexible, elastic 'net' (the 'net' portion isn't really relevant to pasta dough though) , allowing it to stretch more and is much, much less brittle.  Even better, would be to put it through several times, then let it rest for a few minutes so that the gluten relaxes and then continue.   

You could also simply knead the dough longer.  Many pasta doughs are extremely dry and very difficult to knead by hand - so the kneading mainly happens at the roller stage.  If you're using a good mixer or feeling especially hulk-like, then you could knead further by hand.