The EU gives all member nations - and therefore all citizens the right to visit, live or work in whichever other country they want to. There is also the Schengen Zone which consists of European countries (including one or two who aren't EU members) who have abolished all forms of border control between each other. You can literally drive from Germany into Holland without being asked to show anything, and if you missed the sign which says 'Welcome to The Netherlands' the only indication that you'd arrived in a new country would be that the road signs are in a different language.
It's good for both tourism and for business. However it also means that if an EU member country accepts refugees from Syria, for example, then they could feasibly move around freely across the continent. With the ever-increasing air of Islamophobia coupled with the likelyhood of Turkey becoming an EU member state in the coming years this would stretch the EU Border right to Syria itself. You can imagine if you read the Daily Express every day and believe all of the stories of Bulgarians or whatever coming to TUK UR JEAUBS plus the idea that you're about to be blown up by an Allah enthusiast at any given moment you'd want to get shot of the EU sharpish.
- - - Updated - - -
The final graphic in this article lets you compare the EU, Eurozone and Schengen Zone quite nicely.
http://www.economist.com/blogs/graphicdetail/2016/02/graphics-britain-s-referendum-eu-membership