As long as the real rate of interest is higher in the EU, investors will want to park their money in Europe instead of the U.S.. This will require buying euros (and possibly selling dollars) which drives up the price of the euro (in terms of dollars).
Let's look at the strength of the euro in terms of lunch room trades. In this parable let Cheetos = euros and peanuts = dollars. The European Cheeto Bank (ECB) sets the interest rates for Cheetos - if you save one Cheeto today you get two Cheetos next week. Initially a child can trade 5 peanuts for 1 Cheeto.
Suppose on Tuesday, a teen rock idol announces a love for Cheetos (or the ECB raises the Cheeto interest rate), then Cheetos become even more popular. Those unlucky kids whose moms' packed them peanuts will have to give up even more peanuts to get a Cheeto. They might have to give up 10 peanuts for 1 Cheeto. The cost of Cheetos in terms of peanuts has increased from 5 to 10. The Cheeto (euro) is strong against the peanut (dollar).
However, suppose on Thursday, a popular starlet announces that she thinks Cheetos are gross. She further exclaims that anyone who wants to be thin and cool should only eat natural foods, like peanuts (e.g. European banks' balance sheets are revealed to be gross). Peanut packers are suddenly looking good. The cost of Cheetos in terms of peanuts might fall to 4 as students flock to the safety of food without artificial colors. The Cheeto (euro) is weak against the peanut (dollar).
And yes, I did trade peanuts for Cheetos at school. Fortunately, I had a nice friend who gave me Cheetos even though there was no market for peanuts. While this does suggest that economists need to model altruism more carefully, we can probably safely ignore it in the foreign currency markets.