This is my theory.
A large chunk of the press has spent the last couple of decades poking the EU wound with the public, stirring up a victim mentality and pointing people's blame in the wrong direction for decisions taken by our own government, at both national and local levels. The main reason they have done this, it seems, is because most of the big press proprietors see themselves as anti-establishment, small-state Thatcherite barrow boys who still get pissed off at being made to feel arriviste in Britain back in the 80s. The EU is too 'big government' for the likes of Murdoch, Desmond and Dacre. None of them have any levers of influence there.
Meanwhile, there has, since the late 90s, been a cultural movement towards pride in ignorance and lack of questioning. I think a significant locus point for this was Jade Goody on Big Brother but I could be wrong here. Suddenly, it was cool to loudly profess ignorance, to not question one's own thinking. An entire genre of television and online entertainment has grown out of this cultural shift.
It has allowed the atmosphere, stirred by UKIP, Trump and others, where how you feel is more important than what you think. It's then easier to maintain positions that don't stand up to rational analysis. That's why there is so much empty jingoistic nonsense around many of these positions.
Related to the above, immigration is an easy bogeyman for the press to point at because it very easily stirs up people's fear of 'other'. But the reason people can't get doctor's appointments is not because of immigration. It's because since the early 80s the government - of both colours - has squeezed all levels of healthcare funding, meaning surgeries struggle to expand, new doctors can't be hired and new hospitals can't be built.
Immigration is necessary economically and is never going to stop. And neither should it. Countries that are too suspicious of immigration turn into paranoid hermit nations, forever fearing the future. Governments talk a lot of shite about cultural enrichment but that's not why it is done. It is necessary, here, because fewer people are getting married and fewer couples have multiple children. And the sheer size of the baby boomer generation - a population bulge - means that an upswing in pension and healthcare costs needed to be planned for and paid for.
Brexit is a red herring and will not help most of the people that voted for it, as it seems to be primarily designed to foment and attract the kinds of businesses who dislike things like tax, regulations and fair wages and conditions for workers. But my main worry is that the anti-immigrant rhetoric will turn properly racist after Brexit leads to further economic problems, alongside increased immigration from other areas, as organisations can no longer source the resources they need from the EU. Then we really will see a rise in the fascist right.