The crude answer is that people believe what they need to believe, which is not primarily about logic. The assertions that EU membership costs £350 Million per week and that Turkey is about to join the EU bringing mass migration have been thoroughly discredited. The fact that people are still believing them says a great deal for people’s needs and fears. Working with unconscious fears is also core territory for psychoanalysis.
I’ve heard several people recently talk of the “politics of untruth”, as if things have moved into a particularly surreal space in the age of Donald Trump and Boris Johnson. What happens when truth disappears? Are there unconscious things drawing people to vote in a way that makes life worse?