Reading this history/primary source book I found on the street, and in one excerpt, Roosevelt mentions concentration camps as one of the many marks against the Axis, in 1940. To what extent did he know about them then?
Everybody knew about the concentration camps in 1940.
These were not extermination camps. The early concentration camps were prison camps, established to punish and terrorise the regime's political opponents. After Hitler came to power in January 1933, a Dutch communist and recent immigrant to Germany set fire to the Reichstag (parliament) building on February 27 as a protest against the regime. The nazis seized upon this as a pretext to outlaw the communist party and arrest tens of thousands of communists and other left-wing opponents as well as Jews. These were incarcerated in about 100 improvised camps. The best known and one of the only early camps to subsist until the end of the war was Dachau. Most of those early prisoners were released fairly quickly. Some were released because of international pressure, or as in the mass Christmas release of 1933. In 1936 these smaller early camps would be consolidated under the auspices of the SS into a number of large, uniform camps that we are now familiar with, such as Buchenwald, Bergen-Belsen, Sachsenhausen, etc.
Who was sent there and if and when they were released was mainly decided by the Gestapo, the Secret State Police. Some prisoners were sentenced to definite periods. Some prisoners were held for only a short time as a way of terrorising them, for instance the 30,000 Jews that were hauled off to camps in November 1938 following Kristallnacht were all released, some after a few days, some after a few months (except for the 2000 that died...). Those arrests were part of the concerted effort to drive the Jews tp emigrate. Except for this influx in 1938, however, most of the inmates before the war were not Jewish, but political prisoners, homosexuals, Jehova's witnesses, and increasingly common criminals as well.
The existence of these camps was not secret and they were known about abroad and reported on in the press, much like we now know about North Korean prison camps or we knew about the GULAG camps in the Soviet Union during the Cold War.
The camps that nobody was supposed to know about were the death camps (there were only six), and those were established in 1941-1942.