Well, I believe to start it has to be stated that this is somewhat of a pedagogical question, which is not necessarily my strength. However, I am currently writing my dissertation on the Russian nobility on the eve of the revolution, so there is some overlap.

The relatively simple answer is that Rasputin is an easy scapegoat. He is an odd mystical man from Siberia who "magically" helped the Tsarevich's hemophilia. Simultaneously, he ingratiated himself within the royal family. All of this, even during his time, made Rasputin appear bizarre. During WWI, Nicholas II was off playing general (which he utterly failed at) leaving his wife at home with Rasputin. This is where people generally connect Rasputin to the revolution. The story goes, Rasputin influenced Alexandra into ignoring the valid protests of the people thus dooming Russia. In reality, Rasputin barely mattered.

The reality of the revolution is far more complex and layered than one mystic's influence. In order to better appreciate the revolution one has to go back to 1825. In that year, Tsar Alexander I died leaving a confused succession. Alexander had no sons, thereby leaving only his brothers Constantine and Nicholas. Constantine never really wanted the throne but never publicly announced that. Nicholas was too conservative and obsessed with order to just take the throne without Constantine's public abdication. In the meantime, veterans from the Napoleonic Wars desired a more constitutional system of government after seeing Europe during the wars. These men (largely of noble birth) began demonstrating outside the palace in St. Petersburg. Thus, Russia was witnessing a crisis of succession at the same time as having protests for a new government. Nicholas felt threatened (especially when the protestors began marching towards the palace) and called in the guards to open fire. After quelling the Decembrists, Nicholas (who was finally crowned Tsar) began a decades long reign of repression.

A few decades later, in 1853, Nicholas invaded the Ottoman Empire thus launching the Crimean War. Russia was humiliated and Nicholas died shortly thereafter. His son Alexander II inherits and began to modernize the country, culminating in the emancipation of the serfs in 1862. This is when things begin to spiral. Liberals within the country wanted to go further and wanted a constitution. Alexander refused thus leading to increased levels of terrorism. At the same time the nobility just lost a great deal of income during emancipation. In this melieu of confusion Alexander II was assassinated. His son Alexander III was staunchly conservative and fought to stem the terrorists. He did so through secret police and repression. Towards the end of Alexander's reign the country experienced an agrarian crisis. The nobility in an attempt to stave off financial failure, began selling off land. At the same time, the newly freed serfs also had to sell land in order to also avoid collapse. Both groups became increasingly angry at the government.

Alexander III finally passed in 1894 and the throne passed to his son Nicholas II. This Nicholas was unprepared for the throne and wanted, in many ways, to return to a medieval model of government. Nicholas also hid himself away from society and isolated himself in Tsarskoe Selo. With such a lack of contact with his people, Nicholas failed to realize the issues in the country. What he did understand was the need to industrialize. Industrialization is always an interesting topic in Europe but particularly in Russia.

Before we continue, we have to take a step back. Russia despite existing in the modern era, never was quite at the same time level of development as its European peers. Russia was largely agricultural. However, Russia was also existing while ideas of liberal reform, Marxism, and socialism were spreading in Europe. Yet, Russia was a staunchly autocratic country with limited industry. Thus, new radical ideas were spreading in Russia without the preexisting conditions that existed in Europe. In other words, Russia was dealing with complex ideas while not having the ability to support such ideas. In this scenario, the intelligentsia (Russian intellectuals) became increasingly hostile to the government. In some cases, this hostility translated into full-blown acts of terrorism including assassination.

Now back to Russian industrialization at large. Nicholas knew that in order to stay on par with Europe he needed to industrialize Russia. In order to industrialize, one needs to pump great deals of capital into capitalist undertakings. However, in a country that is largely agricultural and just experienced an agricultural depression, those large sums of money could arguably be used in better ways. Nevertheless, Nicholas furthered the industrialization efforts thereby angering poor peasants and struggling nobles. Then in 1904, Nicholas decided to fight Japan. This went horribly wrong, prompting the disaffected peasants to rise up in anger. The nobility largely rallied to the Tsar, but the pressure from the peasants was too much and Nicholas gave in. The Duma was created and Russia technically became a constitutional monarchy (just without the constitution).

Ironically, by creating the Duma, Nicholas doomed himself. Staunch conservatives didn't understand how to support the Tsar while he gave away his own power. Liberals wanted more and the radical left were unimpressed by the reforms. On top of this, Nicholas proceeded to utterly ignore and constantly disband the Duma. Thus anyone wanting change more or less gave up trying to work with the government.

Finally, WWI happens. For a brief moment the Russians rallied behind the flag but after constant defeats the people once again turned from the Tsar. Moreover, at home Russians were struggling under the pressure from the war. Protests erupted and the government responded with violence. Of course that backfired and the government realized that to restore order Nicholas needed to abdicate. Thus, the Russian Revolution began. As you can see, this is a very large, complex, and long tale. Even during the events, nobody truly understood everything. So blame found its way to Rasputin. The odd funny mystic was an easy scapegoat. It was far easier to come up with a grand conspiracy about Rasputin poisoning the minds of the Tsar and his family than addressing the structural weaknesses of the country. All of this is to say, when a simple answer can be found, people tend to gravitate towards it. This new simplified tale of the revolution, has since been passed down. Hence why Rasputin is so focused on.

For sources there are quite a few (apologies for formatting, I'm on mobile): Simon Sebag Montefiore's The Romanovs, Don Rawson's Russian Rightists and the Revolution of 1905, Matthew Rendle's Defenders of the Motherland, Edvard Radzinsky's Alexander II: The Last Great Tsar, Bruce W. Lincoln's, Nicholas I, Thomas Earl Porter and Lawrence W. Lerner's Prince George E. Lvov, Franco Venturi's Roots of Revolution, and Mark D. Steinberg's The Russian Revolution 1905-1921