In a nutshell, the US was founded rather recently (relatively speaking) by people who took up arms to win its independence, maintain its security, and as tools with which to "conquer the wild frontier." Since the role of armed citizens was recognized by the fledgling government they themselves created, it was written into law that those same citizens should not be disarmed by that government. For what is still a substantial portion of the population, that liberty and right continues to be recognized and appreciated, especially as we are able to watch more closely what happens, or has happened, in so many other counties whose governments have disarmed their populations, whether that was done before the USA existed, or since its creation. However, it seems to be less-appreciated within than it once was, though that may be rebounding somewhat.
Self and home defense continues to be a major reason people here acquire their first firearms. This is probably partly due to a perception that violent crime is on the rise, despite the fact that, until only recently, it was actually on the decline, and had been for some time. But easy access to media that allows us to be aware of pretty much anything happening anywhere can make one believe "the world is a dangerous place." This, in turn, sells guns, and some of those people buying them actually find out they enjoy shooting them. It's not unlike buying one's first car solely as a tool to get around in, then deciding to figure out how fast it can really go (and if anything can be done to make it go even faster!)
I can't comment on the "gun culture" within the UK. However, I do know that sport shooting and hunting with firearms are probably every bit as appreciated there as in the US. It also seems, given the intense interest in airgunning there, that people still have a strong curiosity regarding the firearms so many of those airguns are made to replicate. You guys take airgunning to quite a level typically not seen here as much, and markets for them, such as in the UK, inspire their makers to be innovative and to develop high-quality and enjoyable products, a few of which have made it into my hands.
Remember that, unlike what the media (especially media foreign to the US) says, our "gun culture" has nothing to do with violence. The high rate of shootings in the US, when compared to many other countries, is due to a different culture in which people still obtain firearms and use them in that culture (or, should I use the term "sub-culture"?) Incidentally, the US still maintains an annual homicide rate low enough to keep the country out of the "top-25" list of countries in annual murders, at least for 2015 (the last year I checked.)