How did Rasputin gain political influence over the Romanov family and Russian political affairs?

Through Alexandra, Czarina of Russia--Alexandra believed in mystics to tell her of omens or the future.

Rasputin was a mystic or at least posed at one. Alexandra believed he could predict the future and with her hemophiliac son, Alexi, she was in constant worry. She insisted Rasputin stay in the palace as her own personal adviser.