The Revolution is usually known as the Cuban Revolution (not the 1952 Revolution) and is usually dated as beginning in 1953, when a young Fidel Castro attempted to begin an armed uprising by attacking two key military barracks in the Easternmost part of the island (the more famous of the two bases was the Moncada Barracks in Santiago de Cuba).

So the first issue would be that it wasn't that Batista fixed an election, though he had plenty of experience doing that as the man behind the curtain during the late 1930s, but that he had taken power through an illegitimate military coup which had overthrown the Cuban Constitution of 1940. He then proceeded to rule by military decree until he was deposed by the Revolution in 1958 (fleeing just before the capital was taken).

Fidel Castro was a recently graduated lawyer who had been an active member in the very political student union at the University of Havana, known as the FEU (Federacion de Estudiantes Universitarias). By the early 1950s he was set to stand as a candidate for the Partido Revolucionario Cubano - Ortodoxo (Cuban Revolutionary Party - Orthodox, hereafter PRC-O) in a provincial congressional election. Batista's coup put an end to any hopes of getting a seat.

The collapse of the fragile democratic order in Cuba also came on the heels of frustration with the Republic during its first 50 years of existence (it came into existence on May 20th, 1902). Between 1902 and 1933 many of Cuba's leaders had been notoriously corrupt or outright dictators. The 1933 Revolution overthrew the worst of these, Gerardo Machado, along with the original Cuban constitution, which had held the offensive clause known as the Platt Amendment which permitted American troops to intervene wherever, whenever, and for however long Washington saw fit (for the good of the Cuban people, of course /s). The next 7 years were not, however, a return to democracy. The first two years after Machado fell were absolute chaos, with governments lasting months or even just weeks or days. After this initial chaos came a series of puppet governments controlled by Batista until he finally ran himself in 1940 and 'won'. After Batista's presidency, however, Cuba seemed set to finally try to implement the radical promises of the constitution that had been formulated in the late 1930s (the constitution of 1940) as well as return to something like a democratic order.

The problem was that the 1940s was one corrupt inept government after another. By the late 1940s the popular Cuban radio personality, Eduardo Chibas, founded the PRC - O on a platform of cleaning up Cuban politics. The slogan was "verguenza contra dinero", which loosely translates to "[a sense of] shame over [love of] money".

Things continued to devolve as organized crime and gangster like tactics slowly increased their control of Cuban political life.

The young Fidel Castro was fully on board with the reforms that Chibas was looking to implement. That is, until Chibas shot himself, live on Cuban radio. He had been duped by someone into making a claim on the radio which, for once, wasn't actually true. He had doubled down and promised to provide proof. Whatever happened, maybe his 'source' left him in the cold and he finally realized he had been tricked, he attempted to restore his lost honor by firing a pistol at himself on live radio. He died.

Even with his death, the PRC-O was set to take, at the very least, a huge chunk of the available congressional seats in the 1952 elections.

Then the coup happened. Batista, after about a decade outside of Cuban politics (he'd been living a comfortable life abroad), decided to return and despite his best efforts could not get enough support to back a return to power through the ballot box. His solution was to just take power, using Cuba's real problems with corruptions and gangster tactics in politics as his excuse to 'restore order'. He also overturned the 1940 Constitution, which meant that constitutional protections for life, limb, and property just went out the window.

When Fidel and his co-conspirators tried and failed to take two important military bases in 1953, declaring it their intent to restore the 1940 constitution and end Batista's dictatorship, they started a snowball affect. They showed it was possible to stand up to the dictatorship.

Batista's reaction, including torturing and murdering the guerrillas even after they had been caught and had surrendered, turned the rebels into martyrs and heroes. Fidel's brilliant speech at his trial, known by its most popular phrase, History Will Absolve Me (La historia me absolvera) became a huge hit after he slipped it out of prison and had people publish underground editions of it.

From 1953 to about 1955 they were imprisoned in the Prision Modelo (Model Prison), a Panopticon detention center notorious in Cuba, meant to break political prisoners. They were pardoned and freed, finally being allowed to go into exile.

From exile in Mexico Fidel organized an invasion force meant to begin a guerrilla war in the countryside.

In the cities, the example that Fidel and others had given, in addition to the understandable pushback against the brutality of the Batista regime which was quickly accumulating a body count, led to urban guerrilla movements unaffiliated with Fidel's movement taking action. One example is the military branch of the FEU, known as the DEU (Directorio Estudiantil Universitario, or University Student Directory) which is perhaps best understood as analogous to the Weather Underground.

This set the stage for the slow collapse of the Batista regime as its public support, to the extent that it existed, dwindled and its military became demoralized, eventually collapsing after years of constant fighting against the guerrillas. A military which, I should add, was not unanimous in supporting the coup. The so-called "Puros" (pure ones) were very opposed to Batista and this division flared up several times during the regime.

Sources:

-Cuba: Between Reform and Revolution by Louis A Pérez

-The Cuban Drumbeat by Piero Gleijeses

-That Infernal Little Cuban Republic: The United States and the Cuban Revolution by Lars Schoultz

-The Cambridge History of Latin America Volume 7: Latin America since 1930: Mexico, Central America and the Caribbean (the pertinent articles by Louis Perez Jr. and Jorge Dominguez)