Well, first things first - the situation is a genuine clusterfuck indeed. I'm sorry to see the results and I'm sorry that UK has been in such a turmoil the past few days.
I'll admit there are arguments for leaving that make sense to me. For example, both Barroso and Junckler after him are either very bad people, or they have really bad PR, because so far they seem to me (and to many people around me) as if drunk with power or whatever. States speak against the integration? The response - there will be even more integration and you're going to like that or there's the door. That's NOT the European Union as it was intended and that's one of the main reasons people in my country have been talking about leaving EU as well (a complete failure to handle the immigration crisis in any way is a close second). Keep in mind that the EU is an organisation created by states of Europe for the states of Europe. If the states voice their discontent, the EU should care. They don't, it seems. I understand the voters not wanting to have anything to do with that and voting for leave, screw the consequences.
Let me stress that I would vote remain, okay? But I can see how an intelligent person could vote for leave (and not only bigots and idiots).
I also don't like some of the remain rhethorics - being in EU means participating in an international treaty, however nobody can force you to stay there. You don't have to be in the European Union! It definitely isn't your obligation or your duty. You'll probably survive anyway! Honestly, it's not the end of the world. It's just an international organisation, in the end. The people have decided they don't want to participate in that and they're entitled to do so. You should not be blackmailed to stay within an organisation that has moved very far from its roots. By the way, the "woman scorned" is mostly Germany and France anyway. I doubt any of the V4 states is going to be angry with UK for leaving, for example.
I hope this will be the final jolt that brings the reform of the EU that's very sorely needed, otherwise there's going to be some dark years in the near future for all of us, IMHO.
However, I'm also concerned about my own little country - let's admit it, even if it's now breaking from within (or at least it seems so), the UK will prevail. They'll manage and they have the resources and the footing and the whatever to survive even outside of the EU. However, the decision was observed by people in my country... We as a nation (I mean citizens now, not our politicians) have been very eurosceptic ever since the few years after our entry. And UK leaving might make Czechs realise they could leave too ... but being a third tier nation, we have pretty much nowhere else to go, apart from jumping in bed with Russia again (let's see how well that one worked out). That's what I'm afraid of. People in general are always stupid.
Also - people, or the general public, or the cizitens of UK have decided they don't want to stay in the EU. Whatever the reasons were, the heads of EU should now take that as a feedback of sorts and act accordingly. If they fail to do so - and so far it seems it's going to be the case indeed - there's something even more wrong with the EU than I expected.
I would vote remain, but to have another referendum right after just isn't right. You cannot vote again and again until you reach a result you'd like. That's democracy trampled over.
(Let me stress that I'm talking from a viewpoint of a person not living in the UK - I don't have the slightest idea about the social and political nuances - I'm only talking about the fact that a state has left the EU. I'm sorry if I don't make many sense, though.)