Why correlation? Carry trade and risk desire
Basically currencies can be divided into two group, the beta aka'risk-on' currencies and the'safe haven' currencies. Risk currencies are countries where interest rates are set higher than the rates of their counterpart's nation'safe haven' currency'
Large percentage of FX flows are made by the investment side, eg hedge funds. Only a small percentage of FX transactions are created for commercial purposes.
Since all of us are living in a globalized economy right now, these fund managers will want to be on the watch for returns and opportunities abroad.
To accomplish this, they will borrow currency with lower rates and use the money to invest in currencies with high rates. In a prospering world economy, these funds will make a killing not only in the resources they spent in that nation but also amassing higher interest rates and appreciating exchange rate (recall that currency of a nation with higher interest rate is always more powerful than just one lower than it) and from whatever gains from securities they bought in the nation (equities etc)
In a uncertain world economy, carry trade movement will reverse, with fund managers liquidating their holdings in the nation with higher rates and exchange back for the currencies they originally borrowed in reduced rates.
So in our current scenario, Eurozone's interest rate is now at 1.25percent and in US rate is at 0.25percent
In a world where economy is growing, stable and risk appetite is high, EUR/USD will rise. The reverse is true when there are uncertainties. This risk on/off movements moves across the currencies board explaining the correlation. This correlation is not just just between currencies but also with equity indexes as well as bonds.