Explaining the economic outcomes of a particular country at a particular time is extremely hard to do rigorously: whatever explanation you can find there will be another country at a similar time, or the same country at a different time, which had the opposite outcome. (The Republic of Ireland is a particular headache for simple economic development theories.) Luckily, economic growth following defeat in wars is not unique to Japan: Germany and Italy also had economic golden ages post WWII. (As did France.) So we can talk about some commonalities with a bit of confidence that we're not just making spurious correlations. 

Japan's economic growth was not unique to the post WWII era: Japan saw rapid economic growth and industrialisation from the Meiji restoration in the 1850s. Even before the Meiji restoration, Japan was comparatively politically stable, as an island it was relatively protected against outside military invasions but it was also close to other major civilisations, like the UK where the Industrial Revolution started. Japan had a market economy and following the Meiji restoration governments eager to support industrialisation. Similarly, Germany and Italy had started to industrialise in the 19th century and grow their economies. 

So Japan, like my European comparators, had an existing set of knowledge and experience with industrialision that they could fall back on, admist the ruins of WWII. 

On top of that, post WWII, while Japan and Germany's industrial capacity was devastated, the USA and, to a smaller extent, the UK, Canada, Australia and NZ were relatively unaffected wealthy countries, making large markets for Japanese goods which the Japanese could then trade for suppliers of industrial equipment and raw resources, which said wealthy countries were good at producing cheaply compared to earlier centuries.  Increased sea transport and airplanes made trade even cheaper: having rich neighbours near by is a good thing. 

Japan, like other western countries at the time, also benefited from the spread of technologies like the internal combustion engine, building motorways, and electrification, which of course all improved internal markets. 

The Japanese and German governments also pursued market economies, with private property and liberalisation of price controls in the late 1940s, although of course neither government was laissez-faire and the relative impacts of that versus government interventions is still highly debated. 

Finally there was probably a measure of luck. 

We can also compare Japan's economic take off to that of the other Asian tiger economies: Taiwan, South Korea, Hong Kong and Singapore all saw strong growth post WWII, starting from very low bases.

In summary, Japan's economic recovery post WWII is part of a broader pattern of economic take offs during this period, and was also for Japan, like Germany and Italy, to some extent a return to pre-fascist rule economic growth and policies. 

Main Source:

Kenichi Ono, The Economic Development of Japan: The Path Traveled by Japan as a Developing Country,  GRIPS Economic Development Forum, 2006. 

Japan came out of WWII in an excellent position to grow economically. This position resulted from factors internal and external to Japan. The external factors are, to my mind, the more interesting: a former enemy (the USA) becoming invested in Japan's economic success.

Let's start with the internal ones though: Japan had a well educated workforce and a history of industrialization. Without those, the rebuilding would have just been building, and, as your question recognizes, that's not easy to do. Japan additionally had the benefit of its bureaucracy. The people who managed its economy before the war were, by and large, still around and still doing pretty much the same jobs after the war (more on that later). So there was a lot of institutional knowledge left in place despite American attempts to liberalize Japanese government. Finally, and somewhat paradoxically, they didn't have much in the way of natural resources. This meant that reindustrialization was effectively the only strategy available to Japanese elites looking to create a tax base.

So Japan had people who knew what they're doing running things, those people had a motivation to reindustrialize, and there was a history of industrialization and an educated workforce to support industrialization. That's all well and good, but might not have been enough without postwar Japan's geopolitical situation.

In the immediate aftermath of the war, the US planned to remake Japan as part of a broader attempt to liberalize the world order. Alongside the creation of institutions like the UN, World Bank, and IMF, Japan was to be demilitarized and democratized so that it would fit into this new system. In pursuit of this goal, occupation forces wrote the current, extremely liberal, Japanese constitution. It's hard to overemphasize just how far the constitution goes. It's most famous for Article 9, which renounces military force as a tool of the state and forbids maintenance of a military, but I'd recommend taking a look at the entirety of chapter 3 sometime: it makes the American Bill of Rights look downright authoritarian. This attempted remaking of Japanese society by and large ignored the bureaucracy, contributing to its postwar leadership and undermining the goal of democratizing the country.

Soon after the war, however, the US realized that the world was not going to be the liberal paradise it had hoped for. As the Cold War began and Chinese nationalists lost the civil war to the communists, Japan's strategic importance increased. The US began to view Japan less as a strategic threat that needed to be kept in check and more as a potential ally against the USSR. This meant that the US had an interest in Japan's economy. A strong Japan would check Soviet and Chinese influence in Asia in addition to offering a convenient and easily defended military base in Asia.

The treaty of San Fransisco, which officially ended the war in 1951, reflects the American desire to make Japan its ally (the USSR did not sign). Reparations happened, but Japan got a pretty good deal on them, both in terms of the amount that had to be paid and the manner in which the payment was to happen. In addition to explicitly recognizing that Japan couldn't afford to make the countries it invaded whole, the treaty allowed for reparations to be negotiated bilaterally between occupied countries and Japan. Since many of the occupied countries were small, poor, and weak, this put them at a disadvantage and meant a lower price tag for Japan. Contrast this with the Treaty of Versailles ending World War 1, which established a commission that would impose reparations.

Furthermore, the treaty allowed for Japan to pay reparations in kind. So instead of paying the Burmese government cash and then the Burmese spending that to build a dam or a railroad, the Japanese could build the dam or railroad themselves. This meant that the money could be spent in ways that stimulated the Japanese economy. Sure Burma gets a dam or a railroad either way, but if a Japanese company does the building much of that money (along with the experience associated with having built a dam or a railroad) is going to flow right back into the Japanese economy. Japan benefited from the creation of economic linkages too. The Japanese government was effectively able to use reparations to introduce Japanese companies as suppliers to its former colonies, creating a market for its exports.

The reasons for Japan's postwar economic growth are bigger than what I've written here. I hope that I have given a little bit of helpful information on the subject though.

It was the result of reforms and financial policy. The Bank of Japan and government developed a great strategy that gave their businesses a technological edge.

First I’d like to note that there are a lot of answers in this thread that imply or directly state that postwar Japanese growth was just a continuation of pre-war growth. That Japan already had a good system in place and returned to normal.

These arguments are absolutely not correct. During the Taisho and early Showa periods, Japanese economic growth averaged a respectable but not rapid 3 percent a year. After the war, they averaged anywhere between 6 and 10 percent depending on the decade. In short, something changed in Japan after the end of WW2 that allowed the country to grow more rapidly and consistently than any country before it.

During the American occupation, many of the regressive institutions in the Japanese economy were destroyed. SCAP, the occupation authority, deregulated small business and broke up the family held zaibatsu corporate dynasties. Many of SCAP’s other reforms were overturned after the occupation ended, but these two changes were permanent.

Within a decade, massive corporations once again rose in the form of the keiretsu, but the keiretsu were vastly superior to the zaibatsu. Both vertically integrated their production and had internal banking divisions to provide financial support, but they differed in management. Zaibatsu were owned and run by families and shareholders, while keiretsu were beholden to their managers with little influence from shareholders in decision making, when compared to Western corporations.

This was at least partially by government design. The Japanese government established the innovative Ministry of Trade and Industry, which forced a highly regulated banking sector to provide low interest loans to industrial development projects. Meanwhile, it held round table discussions of industrial groups where it negotiated territory and marks share between them before plants were built, in order to minimize domestic competition.

Understanding that the real competition was with foreign producers, the government took this counter intuitive approach to maximize economies of scale. While in a closed market, separating factories geographically and negotiating between numerous keiretsu would have created an oligopoly, in an export-driven market with foreign competition, this approach meant owners could safely invest in large factories without fear of “cutthroat competition” reducing their profits and making the investment uncertain.

This coupled with the guarantee of government backed loans to make investment an extremely safe proposition for the keiretsu. Insulated from shareholders, the keiretsu based their economic planning on interest rates - instead of maximizing short term profits, they invested the maximum that they could and borrowed the maximum that they could until expected profits were less than interest rates.

In economic terms, Japan, through this system, had achieved “theoretical perfection”. In economic theory, general investment in the economy occurs until profits do not exceed interest rates, but the wild card is where investments take place. Japan concentrates it’s lending power in the area where they thought profits would last the longest and have the most number of consumers - in other words, exportable industrial products. In most other countries, investments were branching out into the services sector, which generally produced far less exportable goods, and therefore have a smaller “accessible market”.

What this meant was that it would take far longer for Japanese corporate profits to drop to the level of interest rates than it would for those in any other country. More simply, Japan could continue a high level of investment far longer than any other country. Some very brilliant people figured this out.

Credit was cheap enough in Japan that the government and industry focused on building anew instead of rounding out. The former industrial approach involves tearing down factories and building entirely new ones, giving you a gradual technological edge as entirely new plants can integrate many new changes to technology. The second approach, rounding out, involves replacing only a few machines close to their expiration date, but not all advancements can be integrated. Even a 10% increase in expected output was sometimes enough to justify entirely tearing down a plant for Japanese industrial groups.

As a result, by the 1970s and 1980s, many of Japan’s industrial products were vastly superior to those produced by any other country and Japanese industry was the most advanced in the world.

All of this was enabled by the advent of international free trade in the postwar world. All of Japan’s economic actions contradicted prevailing economic theory at the time, because they tended to assume a closed market. Japan found a loophole by realizing that world markets were more open than ever. Their goal was to gain every possible technological and competitive advantage over a period of several decades by throwing cheap money at any industrial renovation, and ensuring maximum industrial profits by limiting domestic competition. Other countries went through cycles of cheap money, but it was generally directed at the services sector and other less exportable products.

I’ll close by saying that the Japanese miracle permanently changed the fields of economics and economic history. Earlier economic historians (and by this I mean 80s and prior) took an extremely deterministic view on growth, where geography and the past determined how much a country could grow and the government could do little about it in the long term.

Today, there is a general awareness in the field that Central Banks are staffed by bright people who are developing strategies and competing against eachother. Japan proved that geography was far less important than economic strategy and institutions in creating growth.

Takada, Masahiro. Japan’s Economic Miracle.

Takatoshi, Ito. A Miracle in Transition.

Weede, Erich. Comparative Economic Development in China and Japan.

One thing that was not mentioned in the answers here: the Korean War.

In the aftermath of World War II, the US economy had already began gearing down from a war economy and transitioning to a peacetime economy. As a result, the US began scrapping and retiring substantial amounts of war material that it perceived was no longer necessary. In light of the nuclear era, the Air Force argued that with nuclear weapons available, conventional weapons were no longer necessary. As a result, despite heavy political resistance, the US chose to began dismantling the Army and Navy in an effort to reduce the fiscal stress of heavy military spending.

The Korean War was an awkward thorn in this bubble. Suddenly, the US was in desperate need of conventional arms and equipment, which had been dismantled just a few years earlier. The situation was desperate enough that condemned rifles were rescued from scrapyards and tanks abandoned on isolated Pacific islands because they would have been too hard to bring home were reclaimed and refurbished. These served to briefly rearm some of the US troops being dispatched to Korea, but a source of war materials and supplies was desperately needed, and the US wanted to avoid having to re-regear the industrial economy for war once again.

Japan, having an impoverished post-war economy (starvation was not uncommon immediately after the war and post-war Japan's economy was extremely depressed as you compare to post-WWI Germany) proved to be an ideal staging ground for the war economy effort that the US needed to supply its forces in Korea. Japanese war factories were reopened and workers were brought back to work. Japanese industry pre-war had been notoriously reliant on labor rather than machines (as labor was much cheaper in Japan) but because the limited machinery had been effectively destroyed by US bombings, the US exported modern industrial machinery to these Japanese factories, making them far more efficient than they had ever been before. The mass influx of US military investment into Japan proved an economic stimulus: Yoshida Shigeru, the then-Prime Minister of Japan, referred to it as a "gift from the gods."

But the Japanese government understood that the wartime stimulus was limited and that once the war was over the milk and honey would dry up. So while the Korean War began to fizzle out in 1952 and 1953, Japan had already began transitioning its revitalized industry into civilian markets. It successfully transitioned after a brief post-war hiccup, and the combination of cheap Japanese labor with modern American-made machinery and management techniques made Japanese industry-already extremely competitive before the war-into an industrial juggernaut that dominated the world economy up until the 1980s. That was the time of Japan Inc. But that is another story for another time.