I voted Leave because the EU in its current form is already in deep trouble and does not have the structure available to reform itself. This is because it was designed to be impossible to destabilise it from within, by (mostly) French and German people who had observed at first hand the devastation of the second World War, and who had observed the subversion of the democratic German state by the Nazi Party, and who can blame them? However, it is now attempting to govern a far larger and more diverse area, and govern it in a far more 'hands-on' way, and the end result is not really working. Just ask the Greeks...
I've seen some people on this thread repeat the belief that if we'd voted Remain, everything would have stayed the same. But it wouldn't. The explicit aim of the EU is to work for ever closer political, social and economic union.
I found the negative campaigning on both sides very very depressing, ranging from the xenophobic twaddle coming from some of the more ignorant Leavers to the economic scaremongering that Cameron and Osborne fostered. One particularly patronising remark I remember is the warning that a summer holiday in Spain was going to cost at least £300 more if we voted to leave!
I believe our country's parliamentary system can and will step back up to the plate. I have heard some of the remoaners (I hadn't heard this phrase until this thread btw!) worrying about the NHS, education, social rights etc. But the vast majority of our system of social welfare etc was set up well before the EU, and I am quite sure that most if not all the laws that have come in since Maastricht, when John Major signed away a great deal of our autonomy without holding a referendum, would have been set up by our own governments anyway. We also successfully funded research etc. I would imagine that on some aspects of joint policy we will continue to work together, notably in ecological matters. I do think we need to take full control back of our territorial waters though, as the Common Fisheries policies have achieved nothing but devastation for fish stocks.
I believe the divorce process will not be much fun for either us or the rest of Europe, but I also believe that we are all too sensible to ultimately fall out, and a workable solution will be achieved.
I also hope that the European advocates of The Project - the concept of working constantly for ever closer union - will learn some lessons about what can and can't be achieved. An idea that seemed self evident in the immediate aftermath of WW2 is not necessarily the way ahead nearly seventy years later in a fundamentally different world.