The best definition for Fascist Corporatism comes from a political scientist named Philippe Schmitter:

Corporatism can be defined as a system of interest representation in which the constituent units [i.e., social and economic sectors] are organized into a limited number of singular, compulsory, noncompetitive, hierarchically ordered and functionally differentiated categories, recognized or licensed (if not created) by the state and granted a deliberate representational monopoly within their respective categories in exchange for observing certain controls

Now Corporatism isn't rule by corporation, rather it was an attempt by the Fascists to find a third way between Capitalism and Socialism. Part of Fascism's appeal is that put itself forward as a third way. So those who were violently opposed to Socialism/Communism but also distrustful/angry at capitalism could find a home in fascism.

So the idea with corporatism was to solve the issues created by capitalism, but without driving into Socialism/Communism. For example Fascists were utterly opposed to trade unions and Socialism, but they also realized that modern capitalism, with its excesses and flaws created a perfect breeding ground for those movements to pop up. Mussolini outlines this in his doctrine of fascism; which is relatively short and worth a read:

When brought within the orbit of the State, Fascism recognizes the real needs which gave rise to socialism and trade unionism, giving them due weight in the guild or corporative system in which divergent interests are coordinated and harmonized in the unity of the State

So the idea behind corporatism was to weld society together into a single unit where each class and corporation would be represented and work towards the betterment of the Fascist state. It was hoped that by giving workers and corporations representation in the state via corporatism, Fascists could continue avoid socialism while making workers happy enough that they do not form subversive labour unions/socialist leagues/ etc.

especially Germany because I would have imagined that Hitler solidified his powerbase beyond the reach of any corporations by the late 1930s/early 1940s

I'm not hugely certain that Nazi Germany can really be called a corporatist state. Hitler certainly made friends with rich industrialists like the Krupp family, and they were certainly influential because of their vast wealth and industrial factories. But they were never so close to the government like the corporations were in Italy. Nazi Germany never had the equivalent of a "Chamber of Fasces and Corporations" which replaced Italy's parliament. As you can probably tell from the title, it was Italy's attempt to allow corporations representation in government.

I hope this helps.

I have one tiny note to add to the excellent explanation from /u/Sid_Burn.

"Corporatism" has a number of different schools, and did not have a fascist origin. It dates back at least to Pope Leo XIII, who encouraged theological and economic efforts along the broad strokes of what was later coined "corporatism." His commission on the subject began in 1881. Fascists did appropriate the term and did apply their own versions of the idea, but to attribute all things corporatism to fascism is inaccurate. The Catholic idea alone has little in common with either the Italian or German models, not to mention the other conceptions of corporatism.

The easiest way to start in on the Catholic version of the concept is via Leo XIII's Rerum Novarum, which is counted as the beginning of the "social justice encyclicals." (Also, this month next year is the 125 anniversary of the document) On its 40th anniversary in 1931, Pius XI issued Quadragesimo Anno as an update to the original document as well as a proposal for how to economically organize after the shock of the recent global economic downturn. This was seen as a proposal for a way to resist both fascist and marxist conceptions of the economy as well as unrestrained capitalist ideas via worker's guilds and trade unions working with their companies and the state for the good of everyone.

While somewhat popular among Catholics, it must be admitted that Catholic corporatism had a relatively small following overall. The fascist governments of Germany and Italy had little desire to model themselves on the Church's ideas, and Franco's Spain was also unreceptive when it came into power. I just wanted to point out that not all versions of corporatism were fascist.

Excellent question.

Political culture in general is based on certain fundamental beliefs. One of them is whether you believe that the individual human or the "corporate group" (meant in a very broad sense of two or more persons) is the fundamental unit of society. This goes back to the feudal age. People who lived as serfs lived their lives believing that their proper role was in service to the lord who protected them. So the values that were instilled in young people growing up involved service to the estate of which the serf was a part. They were taught from a young age that the group was more important than the individual, and they sincerely believed it. This is known as "tory corporatism." Just to be clear, this political culture (and others) should not be thought of as limited in time or place. There are plenty of "tory corporatists" today, and in varying forms everywhere.

Later, the rise of fascism also depended on this fundamental value, but by that time there was some enlightenment, and some classical liberals and the beginnings of socialism. So in order to achieve a society that operated on corporatist principles, they needed a heavy police enforcement. This is known as "fascist corporatism." A "tory corporatist" culture is a corporatist culture that is ingrained and on-going, whereas a "fascist corporatist" culture is one that requires force to be sustained. If you ever hear speeches by Mussolini, or Hitler, you will notice that they never talk about how their policies will help you individually, they always talk in terms of helping your farm, or your business, or your family, etcetera. That's corporatism. Fascism is a direct and opposing response to the liberal (i.e. classical liberalism) and socialist systems that these rulers don't want interfering with their power. Interestingly, in the US people are not too well informed about this, and just believe that communism and fascism are just synonyms. In reality they are opposites. Whereas the communist takes over private enterprises (corporations), the fascist does not need to do that at all. The corporate interest of for-profit private enterprises is the same as the interests of the fascists.

(There are at least eight types of political culture which can be thought of as describing all political activity: anarchism, oligarchy, tory corporatism, classical liberalism, social liberalism, democratic socialism, Leninist socialism, and fascism. They differ on some fundamental issues, and should not be thought of as limited in time or place. Rather there is a little bit (or a lot) of each of them in many and varied instances.)