8 Primarchs turned traitors and joined Horus in his heresy against the Emperor and his crusade.
The simple, global reason is that they were closer to the Warmaster than the Emperor and when Horus turned against his father, they decide to join him out of loyalty. I don't completely buy it. Sure, loyalty and trust in Horus is part of the explanation, but Primarchs are depicted as extremely intelligent, with a high sense of strategy, tactics and logistic, and couldn't ignore that a rebellion would lead to a violent civil war that turns brothers against brothers and turn the Imperium into a wasteland. But many of them joined Horus without a second thought and seems to have no problem with slaying other Space Marines during Isstvan events.
The reasons for the treason of some Primarchs have been explicitly given in the lore. For example, Lorgar believed that mankind needs faith and was disappointed by the fact that the Emperor refused to be considered as a god. His bitterness grew after the events on Monarch. He instigated the Horus Heresy in the name of Chaos God. Angron hated the Emperor since they met, and Alpharius/Omegon decided to join the Heresy to fulfill the plans of the Cabal.
The main reason for Perturabo's treason is loyalty. At the beginning of Slaves to Darkness, Perturabo is shown to not be happy with what happens to the Traitor legions since the beginning of the Heresy. He is also upset by the high cost his legion has to pay in its war against the Ultramarines. He explains that he accepts to find an unruly Angron only by loyalty to the Warmaster.
But this begs the question: why not staying loyal to the Emperor then? The destruction of Olympia must play a part but I am not sure about the chronology of the destruction of Olympia and the Horus Heresy. The lack of appreciation of his actions during the Crusade made him extremely bitter, but he had no reason to believe that this would change after betraying the Imperium.
Why did Perturabo decide to join Horus in his Heresy?