Here's a definition of critical realism from sociologist Christian Smith:
Critical realism believes in ontological realism, epistemic perspectivalism, and judgmental rationality, all held together. This means that much of reality exists and operates independently of our human awareness of it (ontological realism), that human knowledge about reality is always historically and socially situated and conceptually mediated (epistemic perspectivalism), and that it is nonetheless normally possible for humans over time to improve their knowledge about the real, to adjudicate rival accounts, and so to make justified truth claims about reality (judgmental rationality). All three of these beliefs must go together, critical realism says, to keep (social) science on track. Stated negatively, critical realism rejects ontological antirealism (that “reality” is itself a mind-dependent, human construction), epistemological foundationalism (that a bedrock foundation exists for human knowledge that is indubitably certain and universally binding on all rational persons), and judgmental relativism (that all truth claims are relative and impossible to adjudicate). (To Flourish or Destruct, 13)
The following seems fairly well-accepted these days among social scientists, but it is worth adding:
... critical realism also reconnects facts and values, toward overcoming the standard is/ought divide. The modern propensity to divorce the descriptive and the normative ritualistically appeals to David Hume and against the “naturalistic fallacy.” But Hume has been misread on this point, and moderns have wrongly partitioned the empirical and the moral. Critical realism takes another approach, saying that reality is ultimately of one piece and that while the whole comprises both distinctly factual and normative dimensions, these interpenetrate one another and cannot be fully separable. Our empirically descriptive and analytical observations often naturally entail normative and moral implications. (To Flourish or Destruct, 14)
Smith also maintains a web page How to Learn Critical Realism, which he indicated in his book is "for novices".
As to a possible application of critical realism, consider the problem described by anthropologist Mary Douglas and public policy faculty Steven Ney:
There are several reasons why the contemporary social sciences make the idea of the person stand on its own, without social attributes or moral principles. Emptying the theoretical person of values and emotions is an atheoretical move. We shall see how it is a strategy to avoid threats to objectivity. But in effect it creates an unarticulated space whence theorizing is expelled and there are no words for saying what is going on. No wonder it is difficult for anthropologists to say what they know about other ideas on the nature of persons and other definitions of well-being and poverty. The path of their argument is closed. No one wants to hear about alternative theories of the person, because a theory of persons tends to be heavily prejudiced. It is insulting to be told that your idea about persons is flawed. It is like being told you have misunderstood human beings and morality, too. The context of this argument is always adversarial. (Missing Persons: A Critique of the Personhood in the Social Sciences, 10)
Critical realism's focus on ontological realism is an acknowledgment both of the above problem and the necessity of actively theorizing in that domain. Smith's To Flourish or Destruct is an example of such theorizing. One of the thing Smith mentions is that sociologists in America have had a tendency to ignore or simplify the motivational structure of humans, as if that can be [largely] ignored when studying social phenomena. A reason to downplay motivational structure is that it is not directly observable; to do it justice, one has to posit models which are not purely inferred from sense-data. Critical realism is happy to do this; some of its competitors find this unacceptable.
For causality, the blog post Causal realism and historical explanation will hopefully get you started. It is written by philosopher of social science David Little.
P.S. As regards Kant, Roy Bhaskar said the following:
If philosophy is to be possible (and I want to contend that it is in practice indispensable) then it must follow the Kantian road. But in doing so it must both avoid any commitment to the content of specific theories and recognize the conditional nature of all its results. Moreover it must reject two presuppositions which were central to Kant’s own philosophical project, viz. that in any inquiry of the form ‘what must be the case for φ to be possible?’ the conclusion, X, would be a fact about us and that φ must invariably stand for some universal operation of mind. That is to say, it must reject the idealist and individualist cast into which Kant pressed his own inquiries. (The Possibility of Naturalism, 5)