I'd say that explanation oversimplifies and possibly even reverses the order of some things.

During the revolution, Castro was adamant that he wasn't a communist. And going into the revolution, it's quite possible (even probable) that he wasn't. Castro came from an upper-middle class family, was a lawyer, and actually ran in a Congressional race (in the opposition liberal party) for a brief period that Batista seemed to be liberalizing.

Afterward, Castro because disillusioned with the possibility of peaceful change in Cuba and led an attack on some barracks, for which he was arrested. In his time in jail, he gave his History Will Absolve Me speech to the court, which you can see argues for some pretty mainstream liberal democracy. This speech later served as the political framework for the 26th of July Movement (his revolutionary group).

The US wasn't adamantly opposed to Castro during the revolution, either. Different factions of the government supported either him or Batista, and the official response to the revolution while it was on going was muddled (I believe the rebels even received some provisions from the US, but can't recall the details). The New York Times and other US papers had reporters "embedded" (for lack of a better term) with the rebels, and Castro was often seen as a rather dashing, romantic figure. At this point in time, he was focused on wooing the American public, so it's possible that despite his personal communistic feelings he tailored his arguments to find a sympathetic American audience.

However, leading up to the success of the revolution, many Communists came to join in as well (most notably Che Guevara). The time spent in the impoverished Cuban countryside is also said to be something that moved Castro to embrace communism. So following the revolution, Castro began to appropriate and nationalize industries - BIG RED FLAG to the US. Castro's rhetoric grew more Marxist as well, and this further led the US to distrust him. So eventually Castro reached out to the USSR, and the US began to turn toward counterrevolutionaries to get rid of Castro.

There should also be some discussion of the US's long, complicated and quasi-(and not-so-quasi)imperialist relationship with Cuba, and how that would impact both American and Cuban impressions and decision-making, but this is pretty long as is.

TL;DR - Castro didn't start out communist, but grew more sympathetic to the ideals and less trusting of the US over time, the US was skeptical of him as well, which is what led to the turn to the USSR.