EDIT: There is a famous case of "cheating" - Dora Radjen. There is considerable ambiguity surrounding this case due to the deficiencies of the science of the time and due to time that has passed since the event. In any case, Dora did not win a medal at the 1936 games and it is therefore not germane to the discussion. Those interested may see below comment from /u/EqualSky which contains one article from Der Spiegel that gives some more information.
I cannot locate any other instances of documented cheating by Germany at the 1936 Olympics so what else could have led to the dramatic success of Germany?
Boycotts
There was a strong boycott movement through 1935 with many Jewish and Catholic athletes and their organisations wishing to avoid competing in Nazi Germany. The movement was widespread across many countries (including USA, Great Britain, France, Netherlands and Czechoslovakia) but ultimately fizzled out when the Amateur Athletic Union in the USA elected to compete in the games. While individual athletes did make the decision not to compete, the majority of them followed the lead of their respective country's athletic body and attended the Olympics.
It is worth noting that Russia did not compete, not due to boycott but because they did not compete in any of the Games between 1920 & 1952. Equally notable though is that Germany itself barred domestic Jewish athletes from attending the Games and several medal contenders were caught up in that ban including Gretel Bergmann (high jump) and Erich Seelig (boxing).
Finally, 49 nations did actually attend the games which was, then, the largest number of competing nations of any modern Olympics. It was not due to lack of competition that Germany won the most medals.
Host Nation Advantage
Looking at the history of the 28 modern Olympiads, the host nation has won the most medals on 9 occasions with the breakdown as USA (4), Soviet Union, Great Britain, China, France, Germany each with 1 instance.
Particularly striking is the performance of Great Britain which won a mere 2 medals in 1904 when the games were held in St Louis but skyrocketed to 146 and first place 4 years later in London. It is difficult to do a rigorous statistical analysis on such small sample sizes but it would seem that the host nation does enjoy a strong advantage when it comes to competing in the games.
To contrast, Germany won 20 medals in 1932 and rose to 89 4 years later while USA won 103 in 1932 and slipped to 56. Effectively, German success came at the expense of USA performance although I have to stress that an exact correlation between athletic performance is clearly absurd.
It should be noted, as is common, that Germany fielded the largest team. This is due to a variety of factors but chiefly because it is cheaper for the host nation to send a larger team to each Games. This anomaly has been repeated several times across the modern Olympics and undoubtedly plays a part in the host nation's spike in medals.
National Socialism
Hitler and the Nazi Party viewed the 1936 Games as a chance for a propaganda triumph and spared no effort or expense to showcase the benefits of their regime and philosophy.
Anti-Jewish signs were taken down, the capital was decorated and propaganda was everywhere in the form of posters, magazines and other artwork. The film Olympia directed by Leni Reifenstahl was filmed during this period for release in 1938.
This translated to a significant home crowd advantage for the host nation.
Summary
The take home message is that Germany's performance isn't that exceptional in terms of host nation advantage and can easily be explained in terms of propaganda and somewhat poorer than normal performance by other nations (which may itself be due to some reluctance to compete or the home crowd advantage).
There is no evidence that cheating or boycotting produced a net medal advantage for Germany and their medal count is by no means exceptional in the history of the modern Olympiad.