Essentially, if the US cuts off the freebies (walk-in healthcare at your local emergency room, food stamps, etc.) and prosecute employers for hiring criminal entrants illegally (that is, no more jobs for illegals that pay under-the-table so that they have to contribute some of their earnings to all those freebies), they will leave their country because there is no financial incentive to stay here.
leave their = leave the
I don’t believe Romney will do all that, he is proud of working with the other side, this point of his was a joke and that is why he got laughed at. The absence of clapping made the snickers all the more loud.