This correlation is not easy to explain with traditional macroeconomic models (to some this might not be a surprise given how bad some of our traditional macroeconomic models have helped us navigate through the current financial crisis,...).
To explain exchange rates we normally look for changes in the economic fundamentals of one economy relative to another (the exchange rate is by definition a bilateral variable). If this was just news about the US economy, what is normally assumed (and, under certain assumptions backed by economic theory) is that good news about the US economy get reflected in an appreciation of the currency. But here we have the reverse, the dollar depreciates when the US stock market does well.
Of course, if we truly want to make a relative statement about the US versus Euro economies we are missing information on the Euro countries in the chart. We know that during these months the behavior of these stock markets has been almost identical, influenced by news about the global economy. So, in some sense, it is not clear whether this is a correlation between the US stock market and the exchange rate or between global equity markets and the exchange rate. Should there be a correlation between global equity markets and a currency? It cannot be that all currencies appreciate when equity markets are doing poorly (some currencies need to depreciate!). But the US dollar is not just a regular currency. We are constantly reminded these days that the US dollar is still perceived as the currency of the world and that the only true "safe assets" are assets denominated in US dollars (treasury bills). If that's the case, bad news about the global economy trigger a "flight to quality" and this means moving away from equity but also away from currencies other than the dollar. This is consistent with the picture above although it is unclear how to make this consistent with other mode fundamental theories of the exchange rate (or the stock market) that are relevant at longer horizons -- but in times of panic, who cares about longer horizons?
If this the explanation for the correlation above, then we should expect that as signs of the economic recovery become strong, there will be a corresponding move away from the US dollar, which could lead to a significant depreciation (and more so if the US dollar starts losing its status as the safest currency in the world).