My main source will be "Embracing Defeat" by John Dower; it's the entire explanation of how Japan went from getting bombed by atom bombs to a functional democracy. This kind of question has been asked here before, but Reddit search is abysmal.
The keep myself short, America barges in and took over the already excellent administration of Japan. The bureaucracy that was there was now in US hands in stead of militaristic Japanese. Also, the US brought in western and luxurious goods, food (although too little, famine struck hard), etc. So they were bit by bit converting the Japanese general population that they weren't the bad guys. Using the administration there they also spread their own counterpropaganda to convince the public of their good intentions and that they were not being blamed, only the jingoist and militaristic government who led them to that point. The US coaxed them into thinking it was a cultural revolution for their own good, basically. They also squashed any kind of rebellion or revolution led by others (i.e. communists) very quickly, which also sent a message to anyone thinking of doing the same.
It also helped that the Japanese were good at following orders. So while there certainly was talk of their new "overlords", they did accept the fact that a new sheriff was in town and did their job. However because of every major city being bombed to shreds, millions of soldiers overseas (many of whom would only see home years and years after the war), and very little food, few jobs (factories bombed or closed) the black marked and underground crime skyrocketed which made it a little more difficult for America to control this side of Japanese life.
Later in this process the US gave more and more control back to the Japanese, helped them with democracy with a firm and steady hand, but they never let fully go of the ropes until the mid-fifties I guess it was. And even then they were still monitored closely.
The reason this worked was because it was only the USA that was there, not like in Germany where many different countries competed for the glory. Douglas MacArthur (yes that general) was given free reign in Japan, the public cared little for the Japanese so no prying eyes meant that MacArthur could try whatever he wanted to do. The public though he was going to do the whole "teach a colony how to survive on itself" thing, but because of the excellent bureaucracy in place a new approach was needed. Which obviously worked, if you compare it with certain African ex-colonies nowadays...
Fun fact: While prostitution was illegal, the Japanese government prepared an entire red-light district for the Americans in Tokyo to have their fun in, to "save the Japanese women in the streets who would be raped by the victor". It actually/probably helped as there weren't many incidents of sexual violence by Americans during the occupation. Or at least that's what was reported.