To explain an outcome is to demonstrate what conditions combined to bring it about -- what caused the outcome in the circumstances, or caused it to be more likely to occur. The most fundamental aspect of an explanation is a hypothesis about what caused the circumstance we want to explain. So social explanation requires that we provide accounts of the social causes of social outcomes. (There may be non-social causes of social outcomes -- e.g. "The Roman Empire declined because of the high concentrations of lead in wine vessels"; but these extra-social explanations don't really fall within the social sciences.)
So we need to raise two sorts of questions. First, what kind of thing is a social cause -- how do social facts cause other social facts? And second, what kind of social research can allow us to identify the causes of a social outcome or pattern? (Notice that explaining an outcome is very different from explaining a pattern.)
Generally speaking, a cause is a condition that either necessitates or renders more probable its effect, in a given environment of conditions. (For the philosophers, this means that "C is sufficient in the circumstances to bring about E or to increase the likelihood of E.") Normally a cause is also necessary for the production of its effect -- "if C had not occurreD, E would not have occurred." (The probabilistic version: "If C had not occurred, the likelihood of E would have been lower.) (Wesley Salmon explores the intricacies in much greater detail -- Scientific Explanation and the Causal Structure of the World.)
This account depends upon something that Hume abhorred: the idea of necessity. For natural causes we have a suitable candidate in the form of natural necessity deriving from the laws of nature: "C and the laws of nature => necessarily E." However, there are no "laws of society" that function ontologically like laws of nature. So how can there be "social necessity"? Fortunately, there is an alternative to law-based necessity, in the form of a causal mechanism. A mechanism is a particular configuration of conditions that always leads from one set of conditions to an outcome. Mechanisms bring about specific effects. (So we might say that mechanisms are widgets of natural laws.) For example, "over-grazing of the commons" is a mechanism of resource depletion. And it is the case that, whenever the conditions of the mechanism are satisfied, the result ensues. Moreover, we can reconstruct precisely why this would be true for rationally self-interested actors in the presence of a public good. So we can properly understand a claim for social causation along these lines: "C causes E" means "there is a set of causal mechanisms that convey circumstances including C to circumstances including E."
Are there any social mechanisms? There are many examples. "Collective action problems often cause strikes to fail." "Increasing demand for a good causes prices to rise for the good in a competitive market." "Transportation systems cause shifts of social activity anad habitation."
So this provides an answer to the first question: explaining a social outcome or pattern involves providing an account of the social-causal mechanisms that typically bring it about, or brought it about in specific circumstances.
Now let us turn to inquiry. How would we detect social causation? Fundamentally there are only three ways. We can exploit the mechanism requirement and seek out particular or common social mechanisms. Both social theory and process-tracing can serve us here. Second, we can exploit the probabilistic implications of a causal claim by looking for correlations and conditional probabilities among the conditions associated with hypothesized causal mechanisms. This feature underpins standard "large-N" quantitative research methods in social science. And third, we can exploit the "necessary and sufficient condition" feature by using comparative methods like Mill's methods. In each case, we must keep fully in mind the centrality of causal mechanisms. A discovery of a statistical association between X and Y is suggestive of causation, but we need to be able to hypothesize the mechanism that would underly the association if we are to attribute causation. Likewise, the discovery that a study of multiple cases suggests that A is necessary for E and A&B are sufficient for E requires us to consider the question, what is the concrete social mechanism that links A, B, and E?