As the other answer to this question states, there are almost too many reasons behind Harry's success to list. However, there are a couple of main reasons that contribute to Harry's success that resurface time and again, and are therefore worth focusing on: Harry's personal courage, and his relationship with others.
Though Harry is not the cleverest wizard at Hogwarts (that distinction undoubtedly goes to his friend Hermione), he still manages to repeatedly accomplish heroic feats. While other students are unwilling to challenge three-headed dogs or face fire-breathing dragons, Harry takes on these fearsome challenges with stout bravery. As such, it's clear that one of the most important factors behind Harry's success is the courage that compels him to tackle the frightening adventures his peers shrink from.
That said, Harry would not have been able to overcome Voldemort's evil without a little help from his friends. Indeed, Harry's relationships with others are probably just as important to his success as his courage, and he has help from his loved ones from the beginning of his adventures. Whether it's his mother sacrificing herself to save him, Hermione helping him with his studies, or Dumbledore providing sage advice in times of trouble, Harry is constantly being supported by friends and family. This quality sets Harry apart from Voldemort, who hates others and abuses his followers in order to accomplish his own ends. Thus, the importance of a strong and supportive social network (along with some inner courage, of course) emerges as a main theme in Harry Potter novels.