It is important to remember that Göring was under the spell of a morphine addiction when he was captured by the Americans and was convinced that they would treat him with respect and dignity because he had been a close-aid of Hitler, and was of course, the Reichsmarschall.
At the behest of Eisenhower, however, the American command was to treat all the arrested members of the Nazi hierarchy as prisoners and not men who, despite the grave consequences of their actions, had once held positions of power and privilege in the 3rd Reich. The commandant in charge of the Nuremberg prison was an American colonel*|, who on meeting Göring, stripped him of the large quantity of morphine he had brought with himself and based on the advice of doctors, gradually reduced his dosage of 20 pills a day to the point where Göring would be forced to rehabilitate his drug-addled mind and be fit for the ensuing trials.
After the suicide of Robert Ley, there was a glaring suspicion amongst the Americans that more prisoners would try to commit suicide themselves. Stricter rules were thus enforced.
The most commonly accepted view is that during his time as a prisoner, Göring was able to build influence on the American guard in charge of his cell through his natural charm and charisma now that the effects of his addiction had weaned off. Before getting caught, he had hidden a cyanide capsule inside a jar of cream which he was able to retrieve by asking his cell guard to get it for him, just hours before the hangings were to take place.
After the final verdict that of death-by-hanging, Göring wrote to Eisenhower insisting that instead he should be shot by a firing squad. This was denied and definitely exasperated Göring enough to try and cheat the Americans somehow. His dead body had one eye shut and the other open. For the Americans, it was a bitter realisation that Göring ended up having the last laugh even in death.
*Source: The Nuremberg Diary by Gustave Gilbert. *| Thanks, /u/eternalstudent.