First of all: the Nazi's weren't interested in converting anyone, as they saw Judaism not as a faith but as a species that was inferior to the superior Germanic race. If both your parents were Jewish, then so were you. In fact: the Nuremburg Laws (1935) considered you to be Jewish if you were "3/4 Jewish", meaning that if one of your parents was Jewish, and the other had a Jewish parent, you were considered fully Jewish as well and couldn't get citizenship in the Reich.
They couldn't deny their faith for various reasons, not the least of which was the fact that many were simply betrayed by neighbors or anyone they had a quarrel with.
The most famous example is that of Anne Frank, without a doubt one of the most personal insights into the Holocaust would be her diary.
Aside from rewards being paid for tips about hidden Jews, the Nazi's also uprooted them through fear: early antisemitic laws included the duty to register as a Jew, followed by having to wear external identifiers (like the star of David) and being gathered into the infamous getto's.
In general, the Jewish population cooperated, especially in the very beginning, when the anti-semitism was less lethal and more "in line with the times". We can't forget that antisemitism was widespread in the early 20th century and absolutely not limited to Germany: the USA, for instance, had an increase in antisemitism in the 30s as well, as mass migration of Jews from Eastern Europe and their perceived wealth drove large parts of the population to look envious upon them.
Registering as a Jew may seem foolish in retrospect, as does cooperation of Jewish councils, but from their perspective at the time, it was either emigration (when they still could) or cooperation. We know of extermination camps and planned genocide these days, but imagine you didn't know that: would you uproot your whole family to move to a country that also has its antisemites, or would you just register? From their point of view, what possibly could be dangerous about registering?
So in short: they picked what they thought was the lesser of two evils, and once you were in the system as a Jew, you can't really just go "no, I'm not". In most cases that'd give you a little private time with a member of the Gestapo...
Sources:
If you want to really look into the mindset of the Jewish population before, during and after the Holocaust, I strongly recommend "Maus", by Art Spiegelman. It's a unique biographic graphic novel based on his father's life and really gives a great insight on the whole subject matter.