Or maybe my premise is incorrect and they actually didn't claim this? I just finished Band of Brothers for the first time; while I had heard this before, they made a point of it in the show. Thanks in advance.
Concentration camps were not a secret. The first one, Dachau, opened in 1933 (just shortly after Hitler became Reichskanzler) and prisoners could be sent there, serve their time and then be released.
Even if you mean the extermination camps, none of which were located in Germany (not that the concentration camps in Germany were a good place to be sent as time went on, but at least in those you had a chance of survival), and none were seen in Band of Brothers, rumours still made it back to Germany about their existence due to things such as soldiers on leave or recovering in Germany talking to their family and friends about things that they had seen or heard about at the front.
Germans knew what their government was capable of. Protests against what became known as Aktion T4 (the euthanasia program of those determined to be unfit such as those with mental disabilities) by the public forced the government to cancel (well, mostly) the program. People were really not under any illusions.
See this article from 2013 outlining research done by the US Holocaust Memorial Museum. Bottomline quote: "'You literally could not go anywhere in Germany without running into forced labor camps, P.O.W. camps, concentration camps,” he said. “They were everywhere.'"