Freedom and independence are at the heart of American thought. We became a nation with the government we had (at the time) because we opposed a big government. The state of mind is essentially the following: "It's my property, my decisions and my life, do not tell me what to do."
I think the problem for most people "opposed to background checks" is the idea that, over time, all these small increases in the restrictions on the purchase of firearms will lead to gun law (the only things that can really protect us from a tyrannical government.) and an increase in government with a diminished ability to defend ourselves against those who want to harm us.