I think the fundamental issue is the EU won't allow the UK to take the 'good' bits while avoiding the parts it doesn't want. At the simplest we can't have tariff-free trade without freedom of movement. I'm not sure that was ever explicitly acknowledged before the vote.
Polls show both that a majority of people want a second referendum, but also that if we had one the country would be split down the middle exactly as it was in the first one - it could come out exactly the same or with a tiny percentage victory for remain.
I feel almost as split about it as the country does. I don't think it's surprising that a lot of people are wanting out, because being in doesn't seem to have done much of the country much good. A lot of people are having a hard time, especially outside London and aren't inclined to react well to being told how great the EU is for them. There's the usual ever-present anti-foreigner sentiment, but I don't think that is what has swung the vote at the margins, what did that is the economics. The EU has not been well run and has not made enough effort to take the general populace with it. The UK has also screwed itself over, due to the way it's internal politics interacted with the EU (e.g. successive governments making no effort to use the restrictions on free movement that EU rules already allow for, and opening the borders to the A8 years before the rest of the EU did).
We have a government obsessed with liberalising the economy - making our labour market both easier to get into but also lower-paid and more insecure [_way_ more insecure] - and over the channel are governments struggling with large-scale unemployment and a system that is good for those in work while locking others out of the system. It's not hard to work out how that would play out. It's not all about the awful xenophobia of the British, it's a consequence of different priorities and pressures on different societies, that have interacted in a destructive way. Then you have the Eurozone and the clashing interests of the Germans and the Southern Europeans.
Also Britain doesn't have the same memories of war and occupation, nor the various special reasons for wanting to be in the EU (fear of Russian domination, fear of their own history of fascism, fear of war, etc).
[And it's not insignificant that many Eastern countries are going ultra-authoritarian, the EU is rapidly ceasing to be the club of 'liberal democracies' that it was supposed to be]
But I just can't see that leaving is going to be an improvement, if only because of that tariff-free-trade issue. Plus those in charge of the leaving process appear to be a bunch of clowns and May's government is the weakest in living memory.
Sometimes I think we should just break up the UK, and let the bits that want to stay in the EU stay in while the rest leave, but that's the same sort of despairing thought I have when I look at the political divisions in the US and think of that 'Canusa vs Jesusland' idea. It's not a practical idea.
Probably the least bad outcome would be ending up in some sort of free-trade zone, having to accept almost all of the same rules that we do now, including free movement, but having no vote or say in what those rules are. Which won't satisfy anyone, but having no trade deal at all would be worse.