Ahhh, Rasputin. I have a hard time separating the Rasputin from the movie Anastasia and the real dude, but the real dude was equally if not more crazy (and possibly slightly less of a zombie wizard). This is a VERY basic overview of the topic. There is a lot more going on so anyone else feel free to add you thoughts and expand/correct me.

So, the tsar, Nicholas II, and tsarina, Alexandra, were actually really really good parents. They loved their son Alexei and were extremely attentive to him, which is somewhat surprising because there are lots of stories of kings not being great fathers to all of their children (true? who knows). Alexei was the heir to imperial Russia, which at this time was the last, legit BIG autocracy in Europe. There was certainly still royalty, but most other countries had at least a semblance of a representative government who had power (Russia also had a sort of congress but they had like 1% of the power and the tsar had 99% of the power). When Prince Alexei is a little boy he is diagnosed with hemophilia, which is a terrible disease that almost always kills children who have it at this time (basically it means that your blood won't clot so if you get a little cut you can bleed out).

So we have super attentive and loving parents who are in charge of ALL Russia and the heir to all of Russia is sick with this extremely dangerous illness. In comes Rasputin. Rasputin is from a tiny village in Siberia and when he meets the tsar he is basically wandering around the country as a religious healer and mystic. But remember that Rasputin is straight up a peasant. So the wealthiest and richest people in the country (who have already faced peasant revolts due to lack of food in the 1905 revolts and the terrible Russo Japanese war) invite one of the poorest people in the country to meet with them. Rasputin is allowed to hang around the royals at first as something of an oddity and maybe a party conversation topic, but slowly they start to use him more and more for information about things like "What are the people saying? what is happening in the streets? what do the peasants think?" This is sort of the equivalent of the President of the United States going out and finding a homeless person who is shouting about the end of the world and saying "Hmm, you've got some good ideas, would you like to move into the White House?"

As he starts to get in better with the family he starts to "heal" people. Now I can't speak to how legitimate this is. The doctors at the time can't explain any of his healings, but he definitely does heal some people, including Alexei. There are lots of theories about it now, saying maybe he was hypnotizing people or more likely the doctors were giving Alexei aspirin which was thinning his blood and making him bleed more then Rasputin was making the doctors stop which helped him. The tsarina sees this guy heal her son multiple times. Rasputin starts to become a close confidant of the royal family and lives in the palace with them. So if you are the tsar and the tsarina and you see this guy save your child's life, MULTIPLE TIMES, you start to trust him (not always. Sometimes the tsar gets mad and throws him out but he always come back). And they started to think maybe he does have some mystical powers. And maybe he knows about more than just healing.

So temporary side track, Rasputin was a WEIRD dude. He was really into having sex, in public, with women who weren't his wife (and his wife was usually okay with it). He would hit on girls ALL the time, including the princesses. He was very familial with really important people in Russia, giving them nicknames and basically teasing them. He would run around naked or flash people in public. The newspapers and press loved to write about him because he was the local weird guy who everyone thought was crazy, EXCEPT the tsar and tsarina. He was basically the laughing stock of Russia, but he was in good with the only people who really mattered.

So Russia is deeply involved in WWI at this point on the Eastern front. For a lot of the time, the tsar is away pretty much on the front lines (near the front lines, but definitely involved with the troops) and the tsarina is back in Petrograd (St. Petersburg) running the show. With Rasputin by her side. Many people already do not like the tsarina because she is German and Germany is Russia's main foe in this war. Too ad to this, the poorest in the country have no food to eat because so much is going to the war effort. And the entire time Rasputin has been giving advice to the people in charge about what to do with the country and with the war.

So people are starting to get worried. This guy who seems CRAZY is sitting next to the tsarina, literally whispering in her ear about the future, which is influencing policy decisions all over the country. The British secret service even writes memos about Rasputin and about how they are nervous that he maybe is a German spy or something (remember, Britain and Russia are allies in the war and a crazy man who is maybe a wizard is helping make war decisions). Though the tsar has most of the power, there are bodies in Russia that act as advisers and help actually make policy like the Duma, which was started after the 1905 Revolution to give the people some power in the government. But the tsarina, who is basically ruling the country listens to the Duma, then tells Rasputin everything they say, then Rasputin makes decisions, and the tsarina carries them out.

So if you are a big wig in Russia at this point, be it a legislator or another member of the royal family, you are watching a crazy peasant whose favorite pastime is wagging his dong in ladies' faces as they try to eat lunch RUNNING YOUR COUNTRY INTO THE GROUND. He is making decisions about the war and the troops. Peasants are screaming in the streets about their rights and demanding more food. There are many people in the country who are trying to institute some reforms that probably would have helped quell everything for a little bit, but the emperor and his wife and basically entranced by this man who is saving their child's life and they let him have way more power than he should. Every time someone tries to intervene, Rasputin basically convinces the tsarina that they are trying to sabotage her. The royals already know they stand on rocky ground with the people and have a very delicate position so they don't want to do anything that will upset the balance of power (and by doing that they lead to their own demise). No one trusts the royals because they are being influenced by someone who is a mystical lunatic but they are in ones in charge of the entire country, so there are louder and louder cries every day for change.

I've written a lot so to shorten a great story, they decide to assassinate Rasputin and it takes way more effort than it should. They try to poison him and it doesn't work, then they shoot him and he is okay, then they shoot him AGAIN and supposedly he survives, then he is thrown in the river where he eventually drowns (modern historians think the second time they shoot him probably actually kills him, but it all adds to the mythology of Rasputin.) but at that point it is basically too late and what was set in motion can't be stopped. The Russian Revolution starts in 1917 and imperial Russia crumbles.

Is he solely responsible for the Russian Revolution and the downfall of the monarchy? No, that was probably going to happen eventually. It might not have happened in the same way it did, but these autocracies in Europe were being dissolved. Plus, had the Revolution not started in Russia the outcome of WWI would have been very different which could have meant a loss for Russia, which would have likely lead to government reforms. So I wouldn't say that without Rasputin we would still have a grand tsar of all imperial Russia hanging out today, but he definitely helped push the revolution along and probably lead to the deaths of Nicholas II and his family.

EDIT: Some good books to read for a more in depth look at this: Rasputin: The Untold Story by Joseph T. Fuhrmann A People's Tragedy: The Russian Revolution: 1891–1924 by Oralndo Figes Nicholas and Alexandra: An Intimate Account of the Last of the Romanovs and the Fall of Imperial Russia by Robert Massie

Edit 2: Some spelling/grammar errors

A few quick searches on /r/AskHistorians yielded these results:

Why was Rasputin hated so much in Imperial Russia

Who was Rasputin and how influential was he?