If you commit a felony and are as a result never again allowed to purchase a firearm, your rights are being perpetually limited, and I and a lot of other people object to that, and it's arguably a violation of the 8th ammendment.
If background checks were more limited in the kind of background they checked for, say only restricting people who had injured someone using a firearm, then there might be fewer objections.
And felons who want to buy a gun for nefarious purposes are either smart enough to pay up and get an unregistered one on the black market (so you're not targeting them with background checks) or dumb enough to get one at a local gun store. But the dumb ones are still felons, which means they likely have criminal connections, and if they need a gun they probably can find another way to get one. So by stopping them from buying a firearm legally, they're more likely to end up with unregistered weapon, which makes the DoJ's job harder. I concede that a fair number of them might be inconvenienced enough to just not buy a gun, but then many of them will just go on to stab people instead of shooting them. The number of actual lives saved by background checks is probably much lower than people think.
Further, background checks are notoriously unreliable, turning up plenty of hard-to-resolve false-positives and dangerous false-negatives.
And then there's restrictions on mental health conditions. The problem with these is that people like guns, and if someone who likes guns thinks they might have a mental health disorder like bipolar or schizophrenia that would perclude them from owning one, they won't see a doctor. The result is that they won't get treatment and are more likely to hurt themselves, and indeed, others. Shootings and violent crime have been declining for decades, but the lives of the mentally ill have not been improving. I find the mental health problem much more important than the gun control issues to the proper functioning of our society, it's just that gun violence is more visible. This is a hard problem, and there will have to be compromises but it has to be part of the debate.
I don't know why the NRA opposes background checks, but there are good reasons to.