There were a three main reasons:
The Dutch have a system where all people need to register themselves at their local council, stating the address where they are living. In Dutch, this is called the Bevolkingsregister. At the time of the German occupation, one was required to give many details to the Bevolkingsregister -- one of which was which religion one belonged to. This made it very easy for the occupying German forces to round up the Jewish population or any other 'undesirables'.
We must not forget the large amount of informers which there were. Informers would get food and/or money for turning in people in hiding. As the occupation went on, food became more and more hard to find. It has been suspected that the Frank family was turned in by such informants, but this has never been proven.
The occupying German forces introduced an I.D. card (before and since, there has not been an I.D. card in the Netherlands). A few got caught with fake papers, as mistakes were made in forgeries. The Grüne Polizei (Green Police, named for the uniform they wore) would regularly ask for I.D. papers.
References:
"Leven in bezet Nederland" by W. ten Have (ISBN 9789000344741)