This is my take from the Western side of The Pond. If I'm wrong, I'm sure Simon and/or Davis will set the record straight from their point-of-view.
The EU is a setup designed to unite Europe into a major world market to compete with the US and China, the other two major markets.
To do so, they had to make it look somewhat like the United States. No more passports needed to go from England to France or from Italy to Greece, for example. They also had to get a handle on the rules and regulations that had been the province of each individual country. Standardization of weights and measures, too. This messed with a lot of people's minds at first, but they mostly got used to it.
However, in just about every country, there was an Old Guard who felt that their country's unique identity was being diluted in favor of a global homogeneity of sorts. They also bridled against the loss of independence that was an inevitable part of joining an association like the EU.
In the UK, a lot of these resentments came to a head. A lot of Brits (more than, say, Scots, for example) feel they have a government that pays more attention to immigrants than them. That helps immigrants with government programs that don't benefit the English person on the street, for example.
A lot of those who voted to Exit did so based on misinformation the pro-Exit people fed them, extolling benefits that have turned out to be imaginary. A lot of the pro-Stay people believed polls saying that the UK would vote to stay and figured their vote wasn't needed. Some liberals voted for Exit simply as a vote against the Conservative government of PM David Cameron.
The prototypical Exit voter was older, less educated, white, and male. And also opposed to further immigration, whether driven by racism or economic concerns ("There are few enough jobs as it is. Having to compete with foreigners for a scarce job supply is bad enough, what with it driving down the value of labor. However, to add more and more immigrants to the pool of those wanting work, many of them willing to work for quite a bit less than an English person, is adding insult to injury.")
Another concern related to immigration had to do with terrorism. The borderless EU has obviously facilitated several of the recent European terror attacks.
What those who voted to Exit failed to grasp was the negative impact of the move. The British Pound took a nosedive, which means that every house in the UK is instantly worth less. The value of a paycheck is less.
There's another impact. To exist as a trading nation doing business on the continent, the UK will have to negotiate trade relations all over again, but with little leverage. At the same time, they are likely to have to adhere to the EU's regulations and requirements as part of those deals.
In other words, they will have none of the advantages and all of the disadvantages of belonging to the EU. This is why so many British voters are expressing morning after regrets.
This is a very fair summary.
Free movement is guaranteed in all EU (and Norway/Switzerland). However not all of them are in the Schengen zone. This zone is where passport free travel is possible. It doesn't include England, Ireland, Cyprus, Malta (note those are all Island nations and all of the Island nations in Europe). A couple eastern european countries in the EEA also don't participate in the Schengen because their border security doesn't meet the standards that are expected for admittance in the schengen zone.
Migrants from the Middle East was only an issue inso far as the EU had a rare moment where consensus/unanimity wasn't required to deal with the numbers pouring in. This is the one and only case I know of some countries being forced to accept a percentage of immigrants not from the EU (and for many including the UK a pretty damn small percentage)...against their will. The choice was keep them all in Greece and Austria...or send them back to a war-zone death. They had to be spread throughout the countries...some taking in an enormous amount voluntarily...while the UK took a fraction...kicking and screaming against it. That being said...all EU countries did agree to emergency decision making provisions (which are quite rare) so...they shouldn't complain now if they said it was acceptable before.
Beside that...the UK has (had) total freedom to admit and not admit immigrants from outside Europe. That's not how it was presented by the anti-EU fraudster conmen.
The old-guard comment is true but still to a much more limited extent in other countries. For the most part...the other 26 nations have agreed to increasing integration and growth of the European mission. There is a minority of anti-EU babble in all countries (most notably the Netherlands and France) but it is a minority which could be ignored and is ignored. That's unfortunately not a good idea...as this is why the anti-Europhiles in the UK were able to grow. No one took them seriously or talked back to them...and over years the drip drip drip of their message seemed to enter like hypnosis.
Apart from those small three tweaks...yes...that's a pretty good explanation.
Thanks for the elaborations/corrections.
My understanding is that the EU is a set of trade partnerships and open border policies for its members to use to increase ease of trade with the rest of the world and unite Europe to allow competition with the US and China.
The problem? It takes away a country's ability to police its borders. If one country has a very open immigration policy, it can serve as entry point for all of countries in the EU. To make a comparison you may understand, imagine if California adopted an open border policy and allowed unlimited immigration into the US, once people are in they can travel freely in the US and so they go all over the place: following the jobs and/or welfare.
Another problem: There has been calls for a European army. Doesn't sound like a trading union anymore... Along with the calls for an army, it appears the EU is following a plan laid out in (I think) the 1930s to unite europe into one big country. Like the United States of Europe, for example, which would definitely no longer be a trading union.
Some additional concerns I've heard: It seems like the EU takes legislative power away from individual countries. It's bizarrely complicated, but basically the EU leadership isn't voted into power by the people and the best the people can hope for is to elect someone who will correctly veto the unwanted legislation from the EU and allow any good legislation.
So to sum up: The positives of the EU are that it enables easy trade and travel with/for member countries in europe. The negatives of the EU are that it removes power from the member countries and seems to be moving towards uniting europe into a single country with a non-democratic government.
"imagine if California adopted an open border policy and allowed unlimited immigration into the US."
Already happened, it's called the Southern Border with Mexico.
It was a stupid blunder on the part of PM David Cameron and the British electorate.
The EU claims it only costs about €250 a year per person in the UK. It 'forgot' to add on the national governments price tag for enforcing the EUs rulings. Its red tape and admin etc, etc which multiply this figure drastically.
That would support my suspicion: That perhaps it's possible to have the free trade/free travel without all the bullshit being rammed down your throat. The decrees coming down from Brussels don't seem to have anything to do with you being able to cross borders within an hour's drive. (That would be a novel experience for me, I am two and a half hours (at top speed, pushing the willingness of cops to let me go over the speed limit) from a state line (Kansas, and you'd probably faint at how utterly unpopulated it is there) and Canada and Mexico are both over ten hours away.)
Effectively, based on your description, Europe is no longer a group of representative democracies.
two and a half hours (at top speed...)
How quaint. 5+ hours for me here in South Australia. Then again, Australian states are much larger, geographically, than US states. Don't even get me started about getting into another country!
You can probably get to the edge of Oz pretty quickly (unless you are one of the rare Aussies who doesn't live near the coast). It's driving beyond the edge that would present some difficulty.
But we both have a very different scenario from some parts of Europe, where throwing a stone over you shoulder could cause a diplomatic incident.