The study definitely had everything to do with the compromise of ethics in the face of obedience to authority. And I agree with the last poster that Americans, even today, probably think that, overall, they are more inclined than other cultures to question authority because that is the cliche/talking point about our own democracy. This study questions that assumption.
Another implication of this study is to see the link between culture and obedience to authority - even though repetitions of this experiment tended to end up with similar percentage results. But one example is a chapter from Malcolm Gladwell's Outliers which supposes that certain cultures have a greater instilled obedience to authority. One such example was Korea, where airline pilots were so reticent to question authority that they would candy-coat real problems in the air. Gladwell's contention was that this resulted in airline crashes.
So, Milgram's experiment has other implications; although it was intended to answer the particular question about whether Nazi followers were in fundamental agreement about the genocide or if they were merely following orders, even if following those orders went against their ethical beliefs. In any case, it is troubling. And in any case, it has to do with cultural inscription as well as psychological tendencies. The results of the experiment are chilling in any case at any time, but especially since they were conducted in America and in effort to respond to the psychological "reasoning" of the Nazi mentality during the Holocaust.
The general result of this test was that the test subjects were much more willing than expected to subject the other person in the experiment to what they thought were really high voltage electric shocks.
What this tell us about American society (at least at that time) is that it was much more focused on obedience than we would have thought. It showed that this group of Americans, at least, would be willing to do pretty bad things just because they were told that they had to. This was pretty surprising for a society that had seen itself as so different from Nazi Germany just a couple decades before the experiment was conducted.