The phrase "correlation doesn't imply causation" gets overplayed. (As Cohen wrote, "it's an awfully big hint".) We beat this phrase into students because of a bias intrinsic to the human mind. When you hear 'the crime rate is correlated with the poverty rate', or something like that, you cannot help but think this means that poverty causes the crime. It is natural for people to assume this, because that's the way the mind works. We use the phrase over and over in the hopes of counteracting that. However, once you've absorbed the idea, the phrase loses most of it's value, and it's time to move on to a more sophisticated understanding.
When there is a correlation between two variables, there are two possibilities: it's all a coincidence, or there's some causal pattern at work. Calling a pattern in the world a coincidence is a terrible explanatory framework and should probably be your last resort. That leaves causality. The problem is that we don't know the nature of that causal pattern. It could well be that poverty causes crime, but it could also be that crime causes poverty (e.g., people don't want to live in a high-crime area, so they move out and property values fall, etc.). It could also be that there is some third variable or group of variables that cause both crime and poverty, but that there is, in fact, no direct causal link between crime and poverty (known as the 'common cause' model). This is especially pernicious, because, in a statistical model, all other sources of variation are collapsed into the dependent variable's error term. As a result, the independent variable is correlated with (caused by) the error term, leading to the problem of endogeneity. These problems are very difficult, and should not be taken lightly. Nonetheless, even in this scenario, it is important to recognize that there is real causality at work.
In short, when you see a correlation, you should think that there probably is some sort of causality at play somewhere, but that you don't know the nature of that causal pattern.