There is a few reasons why Rasputin was hated at the time of the Revolution in Russia. However, it is going to need a couple of things to keep in mind.
Grigori Rasputin was a monk from a small village in Siberia. He was brought to St. Petersburg because he allegedly had healing powers and could help the struggling Tsarevich Alexei Romanov (heir to the throne). Alexei suffered from hemophilia, which was a bane of other rulers of Europe thanks to intermarrying of the royalty. Alexei was also the only son to Tsar Nicholas II and therefore the only "true" heir to the throne. Personally, I have never seen why Rasputin was able to "cure" Alexei's affliction or what he exactly did, however that is unimportant to the question ultimately. The point is, Rasputin helped or was believed to help Alexei, according to Tsaritsa Alexandra (wife of Nicholas II).
This belief and relationship becomes important. In 1914 the Great War (WW1) began and Russia entered the war against the Central Powers. The war was not going well for Russia and the Tsar decided he needed to lead his army from the front, and traveled from St. Petersburg to the front lines to lead his army, leaving the Tsarista in charge in St. Petersburg, in 1915.
It was believed, by the nobility in Saint Petersburg, that Rasputin had the Tsarista's ear. That he would give her advice and would help change things to his liking. This belief actually predated the Tsar's leaving of St. Petersburg, however it only intensified after Alexandra had control over the governmental affairs in the capital. Add in a lot of rumors of Alexandra and Rasputin being lovers and you have a mess of a situation occurring in St. Petersburg leading up to the Revolution. Also keep in mind that the war is still going terribly for Russia during the time, strife is happening in the cities and the countryside, and you have revolutionary intelligentsia readying. Anti-war sentiment was also very high. Rasputin became a lightning rod for all of these ills, and the failing Russian autocracy was seen as being corrupted by a monk from Siberia who was allegedly bedding the Tsarista while the Tsar was playing soldier on the Eastern Front. Rasputin then became, in the minds of some, a sort of anti-Christ figure, corrupting the royalty, and bringing strife to Russia.
The anti-Christ sentiment can also be tied to Rasputin's alleged religious sect, which was seen as a corruption and defilement of Orthodox values. He was alleged to be part of a sect called the Khlysts. The Khlysts intertwined sexuality and religion together, while rejecting Orthodoxy. There was also sects of Khysty who used orgies during religious gatherings, or lust and sex in general to become closer to God, by repenting for their sins. These beliefs were very against Russian Orthodoxy and in turn the climate in St. Petersburg and Russia as a whole. Add in there was rumors, on top of the others of his and Alexandra's affair, that Rasputin would bed virgins or young girls while in the capital.
Finally, I will add that there is also a belief in Russia before this that adds to Rasputin hate. There is a belief within the Russian Empire by the people that the Tsar is himself good, or even divine. However it is the corrupt nobility, the corrupt politicians and ministers, that make the Russian government ineffective, cruel, or wrong. That if the Tsar himself saw the plight of his people, he would bless them, however the corrupt politicians prevent him. Rasputin kind of embodies this as well. He is the corrupt one, whispering into the ear of the Tsarista and the Tsar's ears, leading to the problems of Russia. This belief was largely held by the peasants/serfs, however it is worth noting here as well.
Therefore, Rasputin was hated because he was from the outside, he was alleged to be part of a anti-Orthodox sect, and he was seen as the one causing the problems in Russia because of his influence on Alexandra, which was later multiplied when Nicholas II left the capital. Add in some more scandals, and the collapsing Russian Empire, and you have a man that is hated, so much so he was killed.