When we say that Rasputin started the Russian Revolution, we do not mean that he purposely did this or that he wanted to start the Revolution. It is probably better to say that Rasputin helped cause the Revolution rather than to say that he started it since saying that he started it implies that he meant in some way to do so.
Rasputin helped to cause the Revolution because he helped to make people even angrier at the Czar and because he helped to weaken the Czar's government. Rasputin's influence with the Czarina, and through her the Czar, allowed him to have effective control over who got put into high positions in government. This weakened the government because he kept putting incompetent people into power. The people would hear stories about Rasputin and his influence and that would seriously reduce their trust in the Czar.
By reducing people's trust in the Czar, and by weakening the government, Rasputin helped to bring about a situation in which the Revolution could occur even though he himself was not behind it.