I'm reading Kotkin's biography on Stalin and he says that the Bolsheviks basically took power in their revolution because the Provisional Government collapsed. However, it seems Lenin didn't think they had enough power to hold back a German advance on Petrograd which is why they signed Brest-Litovsk. Starvation, banditry and chaos seems to be a key feature before the Civil War and the Russian Army seemed to have collapsed. How did they create a Dictatorship of the Proletariat from the wreckage of this mess? What did they do?
The issue is complicated. Looking at your points one by one:
The Provisional Government (I'm assuming you're talking about the Assembly elected by the Russian people in November of 1917) didn't "collapse". The Bolsheviks dissolved it by decree in January of 1918. Russian democracy lasted a whopping two months.
The new Bolshevik government could not stop a German advance in 1918. The Bolsheviks basically only held the main urban centers of the country. In a way the peace treaty at Brest-Litovsk was an act of desperation which explains why the Bolsheviks gave so much to the Germans.
Soviet hegemony didn't happen overnight. They took power in January 1918 but it would take almost five more years and millions of deaths before they could assert dominance over the territory that one day would become the USSR. In fact...although officially the Russian Civil War ended in 1923 the reality is the Red Army was still fighting rebel forces in Central Asia until 1934.
SOURCE: http://vlib.iue.it/carrie/texts/carrie_books/paksoy-6/cae12.html
SOURCE: Kenez, Peter (1980). "The Ideology of the White Movement". Soviet Studies (32): 58–83.
Two, the Soviets were simply able to muster more popular support than the dozens of disorganized, often erratic, warlords that opposed them. The Soviets promised a brighter future where all would be equal under a State of Soviets ("workers' councils")...the Whites basically promised at best a return to the pre-World War I status quo which was not that great for the vast majority of the citizens of the old Empire.
Three, the Soviets were better at terrorizing people. To be fair, all sides in the Russian Civil War committed large scale atrocities against their opponents, political prisoners and civilians...but the Reds perfected it to a science. Wholesale massacres of senior leadership when they took White Army units prisoner while sparing the rank-and-file soldiers, hostage taking to ensure popular obedience, public executions of landlords and warlords, surgical destruction of both actual and potential political enemies, etc. In a merciless war they were just better at being merciless. This is a quote from Nicolas Werth, a respected French scholar on communism and Soviet history:
"The Bolshevik policy of terror was more systematic, better organized, and targeted at whole social classes. Moreover, it had been thought out and put into practice before the outbreak of the civil war. The White Terror was never systematized in such a fashion. It was almost invariably the work of detachments that were out of control, and taking measures not officially authorized by the military command that was attempting, without much success, to act as a government. If one discounts the pogroms, which Denikin himself condemned, the White Terror most often was a series of reprisals by the police acting as a sort of military counterespionage force. The Cheka and the Troops for the Internal Defense of the Republic were a structured and powerful instrument of repression of a completely different order, which had support at the highest level from the Bolshevik regime"
SOURCE: Nicolas Werth, Karel Bartosek, Jean-Louis Panne, Jean-Louis Margolin, Andrzej Paczkowski, Stephane Courtois, Black Book of Communism: Crimes, Terror, Repression, Harvard University Press, 1999, hardcover, 858 pages, ISBN 0-674-07608-7. Chapter 4: The Red Terror
So in conclusion...the Soviets were able to defeat their enemies because they were better organized, were an united political movement that were able to better articulate a future for Russia and gain popular appeal and were also very, very good at terrorizing those it couldn't convince to support their aims.
As you read on into the biography, you'll find that Kotkin suggests that Soviet influence in the agricultural areas remained poor into the NEP era. Few party members were peasants, and those that were didn't stay in rural areas. Similarly there were few primary party organisations out there.
In a sense the compromise of NEP was at least in part an acknowledgement at the time that total control wasn't possible. Once the compromise is made there is less motivation to clash with the new government for those people on the periphery.