This issue is one part consumer rights, one part protectionism, one part mildly silly government regulation, and one part evil corporatocracy. A little bit of everything! Sure, consumers are interested in knowing where their meat is from. That's a nice thing to know. But is it really a health issue, or an environmental issue? No, it is an issue being pushed by American cattlemen, who hope that the labeling laws will scare consumers off of foreign meat. Is American meat "better," as a whole? Sure, you could make environmental and economic arguments that it is. Of course, there are counterarguments as well. And more to the point, the average consumer shopping for hamburgers will likely not think all of this through. He will, in most cases, either A) buy what's cheapest, in which case he'll stick with the huge conglomerates like Tyson, and ignore the labeling, or B) be scared off by foreign labeling for vague reasons which are not all that well thought through.
So no, meat labeling is not really the issue at the heart of the meat industry (that would be "the evil of factory farming" and "ethical vegetarianism"), but that doesn't mean that Big Meat's motives for fighting labeling laws with their lobbyists are anything less than purely self-interested. From the Wall Street Journal:
In 2012, the 2.22 billion pounds of beef imported in the U.S. made up 7.7% of the total supply. For pork, imports made up 3.3%, and for lamb, 46.4% of the supply came from outside the U.S., according to the USDA... meatpackers are prepared to continue their fight after the [labeling] rules take effect, hoping for a reversal as lawmakers complete the farm bill.
In short, most of the meat you buy is American anyhow, and nobody in the meat industry really cares about the good of the world, so you might as well come down on the side of more disclosure, rather than less, but don't expect too much actual "positive change" to result. This could all be avoided by just eating veggie burgers.