Final answer:
Fidel and Raul Castro were exiled for resisting Fulgencio Batista's US-backed military dictatorship in Cuba, leading to the Cuban Revolution which eventually toppled Batista's regime.
Step-by-step explanation:
The Castro brothers, Fidel and Raul, met Che Guevara in Mexico after being exiled from Cuba for resisting the US-backed military dictatorship of Fulgencio Batista. Their opposition intensified after legal means to challenge Batista's government failed.
As young revolutionaries, they were compelled to organize members of Cuba's working class to rise against Batista's regime. Their movement culminated in the successful Cuban Revolution that began with an unsuccessful attack on July 26, 1953, followed by their capture and eventual release.
After regrouping in Mexico, they returned to Cuba, where, together with Che Guevara and other Cuban exiles, they eventually toppled Batista's government and Fidel Castro took power on January 1, 1959.