“You don’t have to be a member of the EU to trade with it. Switzerland is not in the EU, and it exports more per person to the EU than we do.”
It’s correct that Switzerland—which isn’t a member of the EU—exports more per person to the EU than the UK. The EU imported €20,000 of goods and services per person from Switzerland in 2015, compared to €5,000 per person from the UK.
The figures are a little unreliable, as they can change depending on whether you look at what both countries say they export to the EU, or what the EU says it imports from them.
Switzerland has some access to the EU’s single market, in return for adapting its laws to match EU laws in certain cases. It also makes a financial contribution.