A lot of people neither understood nor cared about the single market and the customs union imo. They understood freedom of movement because they saw the problems schools were facing with children who didn't speak English, they experienced situations where doctor's surgeries and hospital waiting rooms contained a high proportion of people who weren't speaking English. They knew that EU families were being paid tax credits even though only the man of the family was in the UK. They saw EU families being allocated social housing.
Those facts, in my opinion, influenced the vote because they affected people's lives in a way they could see, hear and feel. The other arguments may have been important, but how they related to everyday life wasn't clearly demonstrated. I think people voted on what they felt would improve their lives if the UK left the EU.
It doesn't really matter now, anyway, as the die is cast. If people suffer as a result of Brexit the blame lies with a man who promised a referendum for political reasons and misjudged the situation so badly he ended his own political career when the results of the vote came in. Other politicians then ended their political careers by appearing to ditch democracy completely.
