Guns are important to politicians because of the NRA. They’re either accepting money to be pro-gun or fighting against the pro-gun types by being anti-gun. This has caused it to be a “conservative litmus test” in American politics.
To normal Americans who are pro-gun: usually their stance on guns is symbolic of general conservative views. An attack on gun rights is seen as being symbolic of overreaching government and rampant liberalism by them. Contrary to popular belief, hunting has almost nothing to do with it. The fraction of Americans who hunt is negligible, albeit more than in European countries of course.
Also, of course, it’s a big deal because of our Constitution. Most Americans have some degree of a sense that “once you start messing with the Constitution, nothing is safe.”