Oof, this is a tough question, not least because it seems to presuppose that there is a universally accepted notion of causality among historians (and a way to prove it)! I'll take a crack on it, but keep in mind that this answer is far from comprehensive; still, hopefully it at least gives you some idea of the complexity of the situation. I'll also make the caveat that I'm approaching this from the standpoint of philosophy of history (though philosophy of history as understood by practicing historians), which is somewhat more articulate on the matter than your average journal article. On the other hand, philosophical approaches to the study of history tend to gloss over how causal arguments might differ between subdisciplines, so be aware that this may paint a more homogeneous picture of academic practice than is really the case.

Nineteenth- and early twentieth-century historians

Your question is ostensibly directed towards contemporary historical practice, but let's step back a bit first and look at how older historians used to demonstrate causality; some methods have certainly faded away, but others continue to be used by (at least some) modern academics. To that end, let's look at one of the most systematic "handbooks" for historical practice in the late nineteenth/early twentieth century—Charles-Victor Langlois and Charles Seignobos's Introduction aux études historiques (1897; trans. as Introduction to the Study of History, 1903)—and its approach towards causality.

Langlois and Seignobos distinguish between "transcendental" causes, like divine providence or the unstoppable march of progress, and "scientific" explanations with origins in special branches of history like law or science. At the time, the historical discipline was still working through an identity crisis—an influential group of positivistic historians were trying to demonstrate the scientific nature of history through the discovery of immutable "laws" governing similar sets of circumstances. This led to work on causality through a comparative method—the historian could determine the causes of an event by comparing it to similar phenomena and making a note of shared factors. Or rather, the historian could determine what weren't causes by noting factors unique to each case; as L&S put it, "the concomitance of two facts in several series does not show whether one is the cause of the other, or whether both are joint effects of a single cause." This approach to causality is known as "common sense" logic. Today, also, it is much easier to disprove a cause than to prove one.

So how can one make positive statements about causality? L&S here make a further distinction between the causes of specific events (e.g., the death of Henri II of France) and those of general facts (e.g., processes of industrialization). For the first, they advise relying on written testimonies (a view that few historians today would hold) on the basis that "the observation of causes by the authors of documents is limited to the interconnection of the accidental facts observed by them; these are, in truth, the causes which are known with the greatest certainty." For the second, they advocate a comparative method, but to contemporary processes processes rather than other historical phenomena, since the former can be studied more completely. Here, too, modern historians would either dismiss the methodology or accept it with the severe qualification that one recognize the limits of imposing the past upon the present. (A similar division between "accidental" and "rational" causes, in which the latter can be fruitfully applied to historical generalizations was made by E.H. Carr in his influential What Is History? [1961].)

Another practice of historians of this period which attracted later criticism was what Marc Bloch referred to as the "idol of origins"—the idea that the beginnings of a phenomenon are sufficient to explain its outcome. Bloch and subsequent historians preferred explanations involving what economists sometimes call "path-dependence"—the by now rather commonplace (in historical thought at least) idea that historical phenomena are (1) products of the entire process of their unfolding, rather than just their beginnings; and (2) contingent—that is, they could have turned out differently if any number of conditions weren't met, that these conditions depended upon other conditions, which in turn... and so on. The idea of historical contingency means that causality is a source of eternal debate among historians.

Counterfactual causality

One common method of proving causality—or rather, a family of methods—is through counterfactual analysis. At its most simple, this means imagining how a given event would have turned out minus the factor whose impact you want to investigate. There is an excellent discussion of this method as practiced by historians of the early/mid-twentieth century in Raymond Aron's Introduction à la philosophie d'histoire (1938; trans. as Introduction to the Philosophy of History, 1948). Aron's approach is largely philosophical, as a way to appreciate the impact of a certain factor on the event being studied; he makes no claims to exact measurement.

A more exact methodology of counterfactual argument was proposed by Robert Fogel, though it (like the rest of the cliometric movement in which Fogel was a key figure) has drawn some degree of criticism from traditional historians. In Railroads and American Economic Growth: Essays in Econometric History (1964), Fogel argued that historians could quantitatively judge the causal effects of railroads on American economic growth by applying econometric methods to a hypothetical railroad-less America (which instead used an extensive system of roads and canals) until 1890, and measuring the difference in the real and hypothetical economies. The approach was followed by other cliometricians/New Economic Historians, who were excited by the possibilities of applying deductive logic to the study of the past. One modern critic of the entire cliometric approach, Francesco Boldizzoni, has argued that this type of counterfactualism relies on predetermined assumptions of causality (in Fogel's case, that towns and cities in the US attracted railroads rather than the railroads "making" the urban and semi-urban spaces) and is therefore poorly suited to such arguments.

Other modern approaches to causality

In his Approaches to the Study of History (1994), Michael Stanford distinguishes between four "courses of energy" that inform general causal understanding: (1) pure nature [causality lines up with causality in natural sciences]; (2) nature in the service of man [causality still largely explained in scientific or technological terms, but also as a result of human interaction]; (3) human societies [causal explanations lie "in terms of the arrangements of society rather than the laws of nature]; (4) intention. Historians interact with these four areas in their attempts to describe causes as "departures from the norm" that caused changes over time. This is similar to the comparative method described by L&S above, and must be pursued with the same caveats in mind.

Finally, I want to stress that the chronological coverage I've provided here may give the false impression of a succession of different philosophies of causality in historical practice. In actuality, these methods have at many times (including today) been practiced by scholars in the same period, and often working in the same institution! A somewhat reductive but still very handy division of working historians' philosophies of history is given by Alun Munslow in his Deconstructing History (1997). Munslow's division of historians between "reconstructionist" (*wie es eigentlich gewesen—*the goal of history is to be able to reconstruct the past as it really was), "constructionist" (covering laws and general formulae), and "deconstructionist" (all history is emplotment, and causal arguments are a restriction of causal space rather than a creative statement) philosophies can provide you with an in-depth explanation of some recent trends in historical thought.