It is not completely clear that the Nazi Party as a group really did rely on astrology to any great extent. However, assuming that astrology was important to them, it would have been caused by the fact that Nazism and fascism were very anti-Enlightenment, anti-rational ideologies.
For almost two centuries before the rise of the Nazis, Europe had been dominated by the ideas of the Enlightenment. These were very scientific and rationalistic ideas. They held that human life could be understood through science and that scientific forces dominated the world.
By the late 1800s, however, some had come to rebel against the Enlightenment. They felt that it was a sterile ideology that had no place for human emotion. It did not allow for people to truly be human but rather reduced them to the status of emotionless, hyper-rational beings. People like this fought back by propagating more romantic ideas. These were ideas such as those of nationalism, which in some cases promoted the idea of a mystical connection between members of a national group. To people who rejected science and rationality, astrology came naturally. The idea that the stars influence us is a more romantic and emotional idea and would be likely to appeal to those who rejected the Enlightenment.