Originally the tobacco plant was founded and grown in South America and it happened an estimated 4000 years ago. So no one really knows when it was brought to Cuba, but it was definitely done before the west arrived. It is believed that tobacco was used by ancient civilizations in South America and Cuba for big rituals.
When the Spanish captured most of Latin America they quickly learned of how useful tobacco was and as such started to import it to Europe. From Europe it quickly spread around the world through the Spanish Empire.
By the year 1717, King Philip started growing tobacco in Cuba for the Spanish empire and essentially it has been growing ever since then. Of course the empire was overthrown in 1817 and since then it has been controlled by the Cubans (and most notable of late was Fidel Castro).
Now before the Cuban revolution cigars in 1959 were famous all around the world. As it is believed the quality of soil is better than anywhere in the world. Cuban cigar plant growers have always taken pride in their work so it was a huge part of their economy. This obviously changed dramatically after the revolution.
Just after the revolution Fidel nationalized all industries and most of the best Cigar makers left the country. As of 1962 the USA put an embargo on Cuban cigars. Funny enough because of this embargo Cuban cigar became even more famous as humans always sort after things that are forbidden. Arguably this is how Cuban cigars become so famous as they are today.
Now as of today there is little issues with people getting Cuban cigars so this image of the best cigars in the world may change. However nationally Cuba is still known as one of the best cigar makers in the world and there is pretty much no corner of the world where they cannot be bought!