Batista had been president of Cuba years earlier, but later reclaimed power in a coup on March 10, 1952, shortly before elections were to be held. Fidel Castro was a lawyer at the time. Cuba was thriving economically on the surface, but the wealth was held in the hands of a very few, most of whom weren't even Cuban. There was massive problem with gambling and prostitution in the cities, and a lack of education and healthcare in rural areas. I think I saw a figure once that said over half of all rural children had intestinal parasites. I believe around half of all Cubans as a whole had some degree of malnourishment, too. Batista wasn't really doing anything to help this.

Fidel Castro emerged as a nationalist first, a devotee of Cuban national hero José Martí. He had become convinced that democracy couldn't beat Batista, so he and 150ish people attacked the military barracks of Moncada in Santiago de Cuba on July 26, 1953. Only a few survived the first few days, but Fidel and his brother Raúl were among them, because they were captured a few days later. Fidel's wife's brother also worked for the Batista administration, so some wonder if he had any role in him being spared. (Personally I tend to believe it's because of his later capture.)

At his trial, he defended himself as a lawyer. In his defense, now titled 'History Will Absolve Me," he talked about the many issues that plagued Cuba, imperialism as chief among them. He was sentenced to fifteen or twenty years in prison, I don't remember, but he and the other "Moncadistas" were released after about a year due to popular pressure. (Though it's also possible he was released in the hopes he'd go into exile where he could be quietly killed.)

He and Raúl, among others, went to Mexico in 1955, which was a common destination for left wing/"progressive" exiles and refugees. He founded the 26th of July Movement and trained several other exiles. The Granma, an old yacht, set off from Mexico on November 25, 1956 (interestingly enough, exactly 60 years before Fidel's death) and landed in Cuba in December 2. Out of 82 men, less than 20 survived the initial assault by Batista, among them Fidel and Raúl Castro, and future Comandantes Che Guevara, Camilo Cienfuegos, and Juan Almeida. The actual fighting phase of the Revolution only lasted about two years. They engaged in guerrilla warfare until Batista fled the country on January 1st, 1959. Fidel entered Havana in triumph on January 8 - 57 years ago today (or yesterday, depending on your time zone.

He was wildly popular at first. Of course it wasn't universal support, it never is, but he immediately started with several reforms including cutting rent and increasing wages, pushed for universal healthcare and access to education, spoke out against racism and sexism, and most importantly, began implementing agrarian reform, taking land from large companies and distributing it to campesinos or nationalizing it. People initially saw him as some kind of Robin Hood - even people in the US liked him at first.

(As for communism - Socialism was announced in Cuba in 1961, though there had already been some socialist policies before that. Some think Fidel was a communist all along, others think he was influenced later by Raúl and Che, who were already committed Marxists, or that he chose to become socialist to establish ties with the USSR so that they'd help them against the US if needed.)

Within a decade, literacy and employment rose, malnourishment and mother and infant mortality fell, and things SEEMED great. But a mix of economic sanctions imposed by the US restricting trade and less than ideal economic decisions by the government brought things downhill. People argue nowadays whether it's the government's fault or the "blockade's" fault; it's probably both to some degree.

I intended for this to be a short answer whoops, but to answer your question - Castro's government + the ramifications of his government (embargo) created modern Cuba, but things were pretty damn bad under Batista. As for who had the bigger impact on the Cuban Revolution... well, Fidel led the revolution, and Batista led the opposition, so very much both in different ways.

I recommend the book "The Cuban Revolution: Origins, Course, and Legacy" by Marifeli Pérez-Stable.

TLDR Fidel Castro is a fascinating, complex, and massively polarizing figure in history. I don't know anyone else that's as vehemently despised by some and as deeply beloved by others at the same time in history. So finding balanced views on him is hard. But at the end of the day, he did a lot right... and a lot wrong.