The UK will be trading with countries, but there will be tariffs and the like between the EU and the UK as a result of this move. There are a lot of reasons for the UK vote, but I think at base level the exit vote was fueled by the portion of the lower middle class that feels left behind by globalization. That is being felt worldwide and we see a lot of it in the Trump candidacy. The EU is more comprehensive than NAFTA and the TPP, but it's really about lowering barriers to joint economic activity and encouraging comparative advantage. When the UK leaves, some of those barriers will increase and the advantages of comparative advantage will lessen. The "good" trade deals that Trump talks about and that some who have fueled the Brexit embrace will likely increase consumer prices. That's just the way it is. Further, the British may have to restore some industries that have eroded and that doesn't promote economic efficiency and will also raise prices.
This wasn't about socialism per se. It may have been, working off Runnin's comment, about providing the benefits of the UK social safety net to refugees and immigrants, but Norway never joined the EU and they have one of the most bountiful safety nets in the world. Margaret Thatcher, hardly a hard core socialist, was one of the biggest promoters of greater European integration. England pretty much invented the modern mercantile state and this move runs counter to that.