How did the Fascist/National Socialist economy work?

It didn't.

The basic feature of our economic theory is that we have no theory at all. - Adolf Hitler

National Socialism did not come to power with a defined economic ideology in place. Instead, it was improvised over time, and which economic theory gained favor at any given time depended massively on who had the Fuhrer's ear.

Originally, the National Socialist 25 point programme was quite socialist in nature, and included the following points:

Abolition of unearned (work and labour) incomes. Breaking of debt (interest)-slavery. In consideration of the monstrous sacrifice in property and blood that each war demands of the people, personal enrichment through a war must be designated as a crime against the people. Therefore, we demand the total confiscation of all war profits. We demand the nationalisation of all (previous) associated industries (trusts). We demand a division of profits of all heavy industries.

Over time, however, Hitler drifter to become more and more capitalist. He remarked that true Socialism respected private property, and consisted only of supplying the people with basic needs. Perhaps one line summarized Hitler's flexible views on "socialism" best - "How, as a socialist, can you not be an anti-semite?".

In the early 1930s, with the advent of the Great Depression and an outpouring of support for radical factions such as the Communists and Nazis, big business began to see the Nazi Party as the guarantor of property rights and the lesser of evils. Accordingly, big financiers made massive contributions to the Party - IG Farben, the German consolidated chemical conglomerate - donated 400,000 itself and was the single largest contributor.

Unsurprisingly, when the Nazi Party gained power, they rewarded their financial backers by restricting small enterprise. They ordered that all industries beneath a certain capitalization had to consolidate and merge with larger industrial groups, and forbade the creation of new industries without a minimum amount of capital.

The helmsman of Nazi economic policy between 1934 and 1936 was finance minster Hjalmar Schacht, an academic whose priority was to put Germany "back on the map". This entailed militarization, which Schacht theorized would stimulate the economy, and public works, including the autobahn highway system. This was expensive, which required a high interest rate. This made the plan doubly illegal: first, remilitarization was banned under the Versailles Treaty, and second, in order to raise enough capital in the Depression, Germany had to borrow beyond the legal interest rate limit. Therefore, Schacht chose to fund his projects through a legal fraud known as the MEFO bills. Creating a false company, he sold private sector securities to financiers, that were in reality financed by the government.

Schacht fell out of favor as the German economy recovered and the Fuhrer drifted towards more radical priorities. The 1936-40 period in German economic management might as well be called the Goring period, because in the leadup to war, the party's second in command was the driving visionary of rearmament. In 1936, Goring was put in charge of the Four Year Plan, which intended to boost heavy industry, refining capabilities, and defend industrial production, to make Germany capable of waging a war with minimal dependence on foreign imports. A state company, Goring works, was created, and rapidly became the largest single company in the world. This plan's motivations were not entirely economic, and, as a result, it irked Hitler's allies in industry, who complained. Nevertheless, Hitler revealed his true priority was militarization, and ignored the complaints. During the "Goring period" of the Nazi economy, the state's financial situation fell into utter disrepair as the country started to become badly indebted. The government had to ration consumer goods to stabilize Germany's foreign exchange situation and to divert more resources towards war production.

After the onset of the war, Goring, beset by other responsibilities such as running the Luftwaffe, focused less of his time on the Four Year Plan. From 1940-42, the dominant personality in Nazi economics was labor minister Fritz Todt. Todt's position magnified in importance after the successful invasions of Poland and France, as his ministry was suddenly tasked with managing floods of forced laborers contributed by occupied territories. Over the course of the war, "Organization Todt" managed millions of forced laborers. The influence of Todt on the German war effort was nothing less than disastrous, as he depended too heavily in slave labor, and was reluctant to introduce women into the workforce. More importantly, Todt failed to address the true problem of German production - its qualitative focus, which contrasted with the low survival time of military hardware in the field. This led to Germany under-producing in almost every major category. However, Todt alone was not to blame for this problem, as the Wehrmacht demanded high quality, customized products, and Goring too had ignored this key issue.

Upon Todt's death in 1942, the war economy fell under the management of Armaments Minister Speer. Speer massively increased production in 1943 by streamlining production, improvising where he could not find proper components, and reducing the number of component parts in industry. While German production increased, it still lagged behind the Soviet Union, which at the time was a smaller economy.

Businesses in the Nazi regime were theoretically independent, but in reality were under the thumb of the government and its demands. Nowhere was this clearer than in the American-owned subsidiaries in Germany, which continued to operate during the war. While Ford Germany, for example, was theoretically still owned by the Ford family, its American management had been almost entirely replaced by Germans by government command.

To truly understand the Nazi economy, it is important to understand the Nazi management style, which started from the top down. Hitler applied the Fuhrerprinzip, or "leadership principle" in his affairs. He envisioned the government as a great pyramid, with him at the top. Infighting among his subordinates was not just acceptable, but encouraged, so that the "strongest" would get their way. Thus, Nazi economic policy depended on whoever could win the most internal battles for dominance in economic thinking - Schacht, Goring, Todt, and Speer, in that order. Economic policy varied based on which occupied territory one lived in, or even between the Gaus (party districts) of the Reich. Local officials had a great amount of leeway in how they implemented central directives - what mattered most was edging other competing government departments out of the picture and achieving dominance in implementing your solution.

TLDR: There was no Nazi "economic theory" - they improvised it over time. Hitler sold out to industrialist donors, and his economic policy was taken over by one plenipotentiary after another, who each had a different ouitlook.

Sources:

Rise and Fall of the Third Reich

Preparata, Guido. Hitler's Money: The Bills of Exchange of Schacht and Rearmament in the Third Reich

Ritschl, Albrecht. Deficit Spending in the Nazi Recovery, 1933-1938: A Critical Reassessment

Scherner, Jonas and Streb, Jochen. Was armament minister Albert Speer really responsible for the German “armament miracle” during World War II? New doubts arising from the annual audits of the German aircraft producers.