He did the same thing most modern economies do. Deficit spending.
Hitler and his Finance minister, Hjalmar Schacht, knew that simply printing money to pay for rearmament would lead to inflation, so they decided to re-finance industry by using credit. Schacht and the largest German industrial firms teamed up and issued "Mefo" bills. Mefo was short for "Mettalurgische Forschungsgesellschaft" - Mettalurgical Research corporation.
Simply put, they used these bills instead of money. They were guaranteed by the state and could be exchanged for Cash at the Reichsbank. They also regulated the Mefo bills so each bill issued was tied to a batch of newly produced goods. This way, they were able to avoid inflation. Unfortunately, this eventually led to a huge amount of internal state Debt, which was fine for Hitler. His end goal was to expand his "German Reich" by invading Europe, and debt gave him the opportunity to make everyone see things his way.
To make the most of the opportunities presented by the Mefo Bills, Hitler also instituted other controls. First and foremost, Hitler cracked down on labor unions not affiliated with the Nazis. This was intended to stop unions from advocating for rights and benefits for workers, which allowed Employers and Big Businesses to spend more money on investing to expand their businesses, since they could cut down on their Employees' wages and other assorted benefits. Also, in order to be able to participate in the Mefo scheme, Businesses would have to agree to re-invest most of their earnings (62%) into rearmament and the economy.
In summary, the money came from thin air. Hitler and Schacht issued Mefo bills that were essentially promissory notes/IOUs, which German big businesses used as credit to finance rearmament.
References
https://voxeu.org/article/macroeconomics-germany-forgotten-lesson-hjalmar-schacht
https://cupola.gettysburg.edu/ghj/vol16/iss1/5/
ETA: links and spelling fixed
Fascinating! Thank you for such an extensive in depth answer, I will need to take some time and digest the lnformation in the links... This seems eerily similar to what the US (and possibly EU) are doing now with Bonds, is there a comparison?