Several-fold reasons
1. It's human nature to focus on outlier/minority cases. It's literally how the brain works. You need to be taught not to think 'the one in the million is likely to happen to me'.
So the narrative 'immigrants are stealing our jobs' is easy to believe in because they are the 'dearly minority', even though the numbers say otherwise. Same with how much money we lose to the EU. Same with how much legislative control we lose to the EU. Out loud it sounds substantial but on paper, on the figures, it's really not.
2. News Corp (Rupert Murdoch's news empire: Sky, The Sun, the Daily Mail, etc) have unfathmably huge sway on voting outcomes. Their support has literally dictated who wins or loses any vote for the past 30-35 years, 1:1. They come out with support for a party/figure/outcome, and that party/figure/outcome will win, because they have such a wide readership. One time The Sun even ran a headline 'it was The Sun wot won it' when they span/swayed a whole election.
3. Focus on a particular 'language', and do so forcefully enough, and you will capture people's minds, with or without reason and logic. So combine factors #1 and #2 above and basically you have a massive ubiquitous campaign PR that played to people's fringe fears and forcefully rammed the false logic down their throats
Plus many, many more.
The argument is meant to be that they don't have to live with their decision. I'm 26 and when I'm 60 I'll still be living with the ramifications of this after they're long gone.