I think most people realise that the European level is essential. A large majority of people agree that Europe can provide better answers to certain international problems. For example in Italy, according to the Eurobarometer, people say migration should be dealt with at the European level.
Think about the threat of protectionism and the necessary response to it, climate change or preserving the environment: people believe these things should be dealt with collectively, because that is how we can make our voice heard on a global stage.