The situation with the eurozone is a tidbit complicated if you've got no background knowledge to it, and I'm gonna go to bed in a few minutes, but the worst mistake when introducing it was imo to introduce a common currency and therefore monetary policy, without also harmonising fiscal policy, inflation and wage development. This lead to pretty worrying divergences between north & south Europe in terms of trade balance, unit labor costs, etc., and coupled with the worst financial crisis the world has seen since the 30es and a few specific problems in some of the southern European countries (e.g. corruption, inefficient tax system, tax evasion), has created some huge problems.
Add to that a stupid ass austerity policy during the past few years.
Although a lot of the Anti-European sentiments are simply expressions of a restrengthening of nationalism across Europe, as is sadly only natural when large parts of the population experience impoverishment and worsening economic conditions. That and that they can't differentiate between the EU as a supernational organisation, and their neoliberal and harmful policies that have been propagated through it.