I think using the quesiton about trust in national institutions - "Trust in Government" and "Trust in Parliament" - as proxies for general political disenchantment risks mixing the general distrust of politics with the national vs EU level of politics. Pity they didn't ask a similar question about trust in EU level institutions (though it would probably have been needed to simplify it into "Trust in the EU").

I think it is also noticable that when asked about the UK-EU relationship, the support for an increased role of the EU decreased already in 1996-1998 and support for status quo decreased in 2008-2012 while support for leave increased. So in 2015 we have 65% supporting either leaving or staying but reducing EUs powers compared with 40% in 1992 when the series started. In 2016 these increase to 76% and shift from mostly stay to mostly leave, but much of the groundwork was already there.

Here are their own conclusions:

bsa34_brexit_final.pdf

Social consequences of EU membership
Concerns about the social consequences of EU membership were key in influencing how people voted in the EU referendum.

• 73% of those who are worried about immigration voted Leave, compared with 36% of those who did not identify this as a concern.
  
• 72% of those holding `authoritarian' views voted to leave, compared with 21% of those holding `libertarian' views.
  
• Multivariate analysis found that, for the most part, only items associated with people's sense of national identity and cultural outlook were significantly associated with vote choice.

Dissatisfaction with politics
Dissatisfaction with politics was less important in influencing how people voted in the EU referendum.

• 45% of those who trust government a great deal or tend to trust it voted to leave, compared with 65% of those who distrust it greatly.
  
• However, there was a greater increase in turnout among those with little interest in politics, as compared with the 2015 general election. 43% of those with no interest in politics voted in the EU referendum, up from 30% in the 2015 general election. Nevertheless, these proportions remain markedly lower than those for people with a "great deal" (90% in 2016
  

and 88% in 2015) or "quite a lot" of interest in politics (89% and 86%).

I can't find the appendix where the details of the multivariate analysis is supposed to be found.

The article, "Brexit: People voted to leave EU because they feared immigration, major survey finds", doesn't define "social consequences" that Leavers associate with the free movement of people into the UK. Apart from national pride and purity might the majority who sanctioned [!] BREXIT and sanctioned [!] Tory austerity fear further competition with immigrants for dwindling "social services" such as universal credit, bedroom taxes, minimum-wage work in construction, and so forth?

I noted with interest two related articles. Here housing market profiteers correlate rising costs in Austria with a forthcoming wave of immigrants, in particular "destitute asylum seekers" (Guardian, Jan 2017) from Britain, Africa, and the Greater Middle East. After accepting 90K of these, Austria's re-location exemption expires September 2017

Property prices around 39 percent higher than six years ago

A new house in Austria (123m2) now costs on average €359,000.
[...]
Bargain prices for older properties can be found in Burgenland, Carinthia and Styria - with average prices for a house between €230,000 and €290,000.
[...]
Rental prices for new properties have risen steeply, by 21 percent, with average costs now €11,50 per square metre. In Vienna council-owned properties, where rents tend to be cheaper and only rise according to inflation, are in high demand. ..."There are not enough properties available and this problem will increase. Around 300,000 people are expected to move to Vienna over the next few years... and the city has not prepared for this."

'Huge gaps' in Vienna housing market

The CEO of property management company Buwog, Andreas Holler, told a press conference that not enough homes are being built and urged the city government to devote more time and money to planning new developments. Vienna's population has grown by 150,578 people since 2010 - and just last year it grew by another 43,236 people, largely due to increased immigration and the refugee crisis.
[...]
The average household in Vienna has just two people.

Can UK's Leaver's be persuaded that a similar opportunity to cash in their equity awaits the arrival of immigrants?

Diversity is the key to economic and political evolution.