Pretty much anyone with knowledge and understanding of the subject agrees that the UK has benefited from EU membership, and will almost certainly be hurt by leaving. (The rhetoric about how much the country pays to the union misses the point completely. It's like arguing that anyone who pays taxes would be better off on their own, even if that would mean giving up access to schools, hospitals, roads, running water, etc.) So this was a stupid decision. No one pretends that the EU is perfect, and dissatisfaction with it is understandable. But the EU's dysfunctions are a reflection of European squabbling, and you can't escape from those problems: You can leave the EU, but you can't leave Europe.
Of course, there are countries that manage outside of the EU, and the UK is in pretty good shape overall (not like Greece, for example), so the consequences most likely won't be apocalyptic. But in lots of small ways that add up, Brits will be worse off. For example, 10% of UK tax revenue comes from the financial services industry, and London has become a hub for this in large part because it's possible to trade all over Europe from there. Much of that business may now (gradually) be lost to other European centers, with knock-on effects for other parts of the economy. So that means loss of growth, recession, deficits, swingeing budget cuts, austerity...
I'm most worried for all the people I know who come from various countries in Europe and live in Britain. This referendum has been a giant "screw you!" to them. And many of my friends (Brits and others living in Britain) are researchers: these days, EU grants is one of the main sources of research funding (though the European Research Council), and if the UK cuts itself off from that it will make international research collaborations far more difficult, and British universities much less attractive for top-quality talent.
And sure, a lot of the potential problems can be mitigated: you don't have to kick out foreigners, you can still ask to join the ERC (for a fee), it might even be possible to remain part of the common market so you don't drive away business. But avoiding the most serious consequences of Brexit means, in each case, adopting the very EU policies and regulation the Leavers were protesting against, and now having no direct influence on them. Great win for "sovereignty", eh?
And that's not even taking into account the possible domino effect this decision may have set in motion, that could lead to the breakup of the United Kingdom. Congratulations on that!
The only way I can think of that this makes sense is if you think the EU is doomed anyway, and that it's better to get out now in a halfway orderly fashion than to be caught up in its collapse. Though if that comes true, it will be the very definition of a self-fulfilling prophecy.