I'm Canadian, not British, but at least traditionally, we have a lot more guns than people do in the UK, so maybe I get it somewhat.
Fundamentally, Americans think that if their government ever became tyrannical, or they were in a position where they needed to fight for their survival as a people, they have the right to do so with dangerous weapons. I think it's pretty clear how this view came out of their history, but it isn't so strange really even to others - in countries that lie close beside hostile nations, people also value the right to keep arms close at hand.
I suppose maybe the best way for a Brit to think of this is to ask yourself - if your government went crazy and started locking up citizens or something, using the army as a tool, would you think it was within the rights of your citizens to take up arms against them?
If you think it would be, then you have in a basic way the same understanding as Americans on this. If you don't, then you believe that citizens should just sit back and take whatever shit an evil government imposes. Or that in an invasion, it is only up to the military to respond, citizens outside the military have no right to defend themselves.
Now, from there what I would say is that to Americans, the right to be able to fight against an oppressive government with arms is meaningless if citizens don't actually have access to them and the ability to use them.
So while they do almost all think there can be some legitimate restrictions or rules around guns, they see the balance as being on the right of citizens to access them so in many states they set a high bar for the government to regulate.
They might be inclined to say that Brits are only so blase about such rights because they are complacent about the possibility of failure of the state or invasion.
Of course this leaves the question of mass shootings, gun crime, and the rest. Many Americans may in fact agree that some gun laws might be better formed. However, they will also tend to say that the real problem is not the right of citizens to access guns (and after all, Americans are not the only people who keep guns at home.) The problems are things like gang culture, the drug trade, depression, inadequate policing, and the weird culture that has grown up since Columbine that sees mass shootings as a way to make a kind of ultimate statement.