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.