To understand why Plato thought that poets were so dangerous it is necessary to consider how Plato viewed the soul of man. According to Plato, the soul contains three parts: the rational part (the part that gives us the ability to calculate and reason), the appetitive part (emotions and appetites that motivate us to engage in certain activities, such as our sex drive), and finally the "spirited" part of the soul. Plato compares this part of our soul to a beast because it represents the instinctive, bestial desire for survival that emerges when we are facing severe danger. Perhaps it can be likened to the "fight or flight" response that we talk about in modern psychology.
Plato, when thinking of these three elements of the soul, believed that to be fully human it was necessary to suppress the bestial nature and appetitive nature of our soul and keep them under the control of the rational part of the soul. However, poets, according to Plato, work by arousing raw emotions from deep within us. Their talent does not lie on proving arguments, but by provoking emotional responses that cause us to act irrationally. Because of their ability to arouse the appetitive part of our soul, and give it more power than the rational part of the soul, he thought they should be banished from the perfect society.