The European Union consists of two parts: an economic union and a political union. The economic union has a single market with the same rules applying to all member states. Non-members of the EU who export to the EU must comply with those rules for their goods, but not for goods they trade outside the EU. The economic union features a common customs tariff for goods entering the EU, but none on trade between EU members. The political union has an elected Parliament with some legislative and budgetary powers; so does the Council of Ministers, composed of 28 ministers (one per member state). The European Council, composed of the 28 heads of state or government defines the EU's policy agenda, and the European Commission is the executive, dealing with the day-to-day running of the EU and enforcing its treaties. It has one commissioner appointed by each state. The aim of the political union is "ever closer union," increasingly giving the EU the characteristics of a unitary state. The aim of many of its adherents is a European state with its own army, embassies, and laws.
Several states outside the EU have negotiated trade deals with it giving them access to the economic union's single market, without becoming part of its political union. If the UK were to leave the EU's political union, it would certainly do the same, since this would be hugely to the advantage of both the UK and the EU. Similarly several non-EU states have agreed visa-free travel with the EU, though of course non-members have to pass through passport and customs controls, as UK travellers currently do.
The case for the UK's leaving the EU amounts to advocating that the UK retains the advantages of the economic union, with access to the single market, and with EU access to its own market, while withdrawing from the political union that gives EU bodies powers over it.
The case for remaining in the EU ought to be about continuing to be a part of a developing political entity, pointing to any gains that derive from this. Instead it has thus far been almost entirely based on the fear of adverse economic consequences, many of them based on the utterly false assumption that the UK would not negotiate an agreement that kept it closely integrated with the EU's economic union.
The UK's citizenry has never wholeheartedly endorsed the political union. When it voted in the 1975 referendum to endorse the legislative decision to join, the arguments were almost entirely about the economic advantages of doing so. The case for remaining in the political union has never really been put to the test, and is not now being done. A vote to leave would be one to reject the political union, while remaining closely integrated with the EU's economic union.