There is an obvious correlation between GDP and democracy. Just think about OECD countries, all democratic, which are much richer that Third World economies, which are often not democratic. This correlation has often been a motivation for imposing democracy on some unsuspecting country, arguing it would improve its economy. But correlation is not causation. And even if there is causation, it could go the other way.
Daron Acemoglu, Simon Johnson, James Robinson and Pierre Yared use two strategies to try and find causality: First introducing country fixed-effects in a panel regression, which are supposed to take into account country-specific effect impacting jointly income and democracy. Second, use an instrumental variable approach, which gives sources of exogenous variation useful for estimating causation. In the first case, the correlation disappears, and in the second, no causal effect is found from income to democracy.
In the latter case, one can discuss forever whether the instruments are adequate, especially as their strength is not reported. Also, causation could actually go the other way (as advocates of imposing democracy onto other countries like to argue). Still it remains a puzzle why rich countries are democratic. The authors advance a few vague ideas (political and historical accidents, long term effects not identifiable in the short sample), but clearly much remains to be done here.