I voted for Brexit, but I did not vote based on immigration policy. In addition to seeing no major problems with the current immigration arrangements from an economic perpective, one of my grandfathers was an immigrant to the UK, and I'm glad to have my mixed ancestry, if nothing else it made my childhood more interesting.
The problem really is with the media. They've set out Brexit as a two-issue debate. The only two issues that get discussed are immigration and the economy. So if you voted for Brexit you're either a racist or an economic luddite. I bet you can't even guess why I'd vote for Brexit without being driven by those two factors. That's the level to which the debate has been simplified in the media.
So let me say this, speaking as a Brexit voter, you are welcome in this country. Don't let the media tell you why those 50% of voters (which wasn't 50% the UK population) voted the way they did, they haven't got a clue.
If you are curious about the reasons I voted to leave, look through my comment history around the time of the Brexit vote. I may explain it again later once I've made my point about the media's role in limiting the public debate.
This position was widely discussed in the British media [1]. Suggesting otherwise is bananas :-D
[1] A Google search for "Brexit sovereignty" gives plenty of articles, from across all newspapers.
One of the complaints I have about how the media has portrayed the issues is that it's conflated membership of the EU with being European, by focusing so heavily on the immigration angle. I'm still a European if I want to be outside of the EU. The EU is still mostly a trade organisation. Nobody calls people from the US anti-American if they criticise NAFTA (they are of course different arrangements, but certain comparisons can be made). Yet the portrayal of a European who is critical of the EU is of someone who is anti-European.
If you want a view that certainly wasn't emphasised in the Brexit debate, here are some of the comments of Tony Benn about the EU:
A rapidly changing first-world is producing a lot of losers - but they're not yet disenfranchised and many of them are even armed.
We're entering a new industrial revolution - a lot of white collar jobs and blue collar jobs are being disrupted or eliminated. For example every factory and driving job will disappear in our lifetimes, many office jobs will be disrupted. New jobs will be found, we'll be more productive, and the world will be better because of it, but history shows us that this sort of disruption and destroyed livelihoods leads to violence, revolutions and unrest. We will see a similar situation to the original industrial revolution.
I do think it is misguided and harmful to attempt to try to find scapegoats for structural changes in society (popular just now are immigrants, muslims, corporations or evil federal governments), which is why I disagree fundamentally with almost every argument advanced for Brexit (from foreigners are taking our jobs to the evil EU is out to crush our democracy). It will not end well when people don't see improvement in their life but suffer more, and wonder who is to blame if not immigrants.