He was born the son of Russian peasants, as many millions were in the course of that country's history. His early life appears to have been reasonably unexceptional if you discount the decision not to become a monk and the scandal of fathering illegitimate children (I'm not sure which decision came first!).
The exceptional thing about Rasputin was his claim to have mystical healing powers. It is at this point that the story begins to feel unreal. It's sometimes like teaching a fairy tale.
(Once upon a time...) Rasputin caught the attention of the Tsar and Tsarina (Russian royal family) and they showed particular interest in his ability to heal their son, Alexis who had haemophilia, a disease where blood does not clot. This mystical ability ensured Rasputin's place at court and he continued to have influence over the affairs of government. This increased when the Tsar took charge of the Russian war effort.