Well speaking as someone who actually did vote to leave, but believes the campaign on both sides has got waaay out of hand I can definitely come at this from a different angle.
First, it is not the case that the question is just about refugees and migrants, for all that a lot of right wing idiots believe it to be, though I will note that as someone trying! to marry an American and sort her legitimate entrance into the country I do find the current immigration situation more than a little silly, given that despite the fact my lady comes from a country britain has had exceptional relations with in the past ie, the united states, speaks English and has no dodgy leanings she is subject to a massive inquisition and ridiculously high immigration charges, where as were she coming from any obscure corner of europe for whatever reason there wouldn't be a problem, and yet there are! some fairly dodgy people coming into the country for far less good reasons without any scrutiny at all, (which I hasen to add certainly doesn't mean everybody from Europe is dodgy, just that the attention given seems extremely unequal).
The issue for me has always been one of administration. A lot of random laws have been passed by the european parliament that just plane werent' in the interests of britain or didn't take into account situational changes over here.
For example health and safety regulation such as energy saving lightbulbs being manditory (despite the fact they actually make it so dim it's hard to see for someone who's visually impared), and a lot of trading standards stuff which make it far more economically difficult for smaller businesses in Britain to function.
For example, a friend of mine was the son of a pig farmer, and had a major beef (ha ha), with the fact that due to Eu trade regulations and the fact that the government was committed to lowering import duties, pork farmers in britain had no way of selling the meat they produced as compared to people in Spain who used extremely inhumane and cruel farming measures that would be illegal in this country.
That's not to speak of other issues, such as the problem of steel works being shut down, and far too many nationalized British services being sold of to european investers (it's fairly nuts that the national health service is currently largely administrated from Spain), rather than keeping things in the economy in Britain.
All that being said, I do think both sides of the debate have got far too nutty and over emotional and inclined to vote with the guts rather than the brain, and indeed the media currently, and trump, and everyone else proclaiming doom and gloom and the end of all things, after all Britain never did! have the euro anyway, and nobody ever said that Britain would stop trading! in europe, only that administration wise Britain was no longer part of the european union as a legislative entity, (nobody ever objected to when it was the e e c).
As to why trump and everyone else has got on the bandwagon, I suspect that is just an attempt to spin current events to whatever political advantage is needed, after all there is no logical reason why Britain's relationship with Europe will actually affect the relationship with the Usa, indeed if anything I suspect Britain and the Us will become politically closer, given common language and frequently common interests.
I imagine for all practical purposes the split from europe won't have that much affect after the initial "oh doom!" has faded out, just more self administration for Britain, and hopefully a bit more of a move back in the direction of nationalisation and against all this placate the big business crap which dominates politics at the moment, though i am a little concerned about whether it will be a platform for the nationalistic right wing nutjobs to gain more power in the country, which is something I'm definitely! against, since I don't imagine having Ukip, ie, the uk independence party in power would be a particularly good thing for anyone.
I know it may surprise people that me, probably the least patriotic person in the world voted in favour of leaving europe (my dad has the ability to vote for me by post hence why I voted against even though I'm currently in pencilvania), but for me it was mainly a question of administrative power and local interest, particularly given my views on multinational capitalism.
As I said, I doubt severely whether things will be as doom laden as much of the media predicts, after all similar predictions were made back during the recession in 2008, but we'll see.
With our dreaming and singing, Ceaseless and sorrowless we! The glory about us clinging Of the glorious futures we see,
Our souls with high music ringing; O men! It must ever be
That we dwell in our dreaming and singing, A little apart from ye. (Arthur O'Shaughnessy 1873.)