I'll start with the why.
The reason that there are so many Latin American baseball players is manifold. It's part socioeconomic, part lack of professional sports opportunities in their home countries and it's part the amount of money, time and effort various Major League Baseball (MLB) organizations have put into recruiting and signing Latin born players. Similar efforts are going on in Korea and other Asian countries, however they have not met with the same success, probably because there is not quite the same rich history of baseball tradition in those countries (although it's coming)
Let's start with the socioeconomic/lack of pro-sports opportunity issue. In most places in Latin America, if you want to play pro sports you play soccer (futbol). It's the only major organized professional sport in several countries, and the only one besides baseball in most of the others. However, the salaries are not like professional sports in the US or Europe, and while they are a decent living, most truly amazing players in soccer or in baseball quickly find themselves whisked away to better competition in the US or Europe.
While Latin America is progressing economically, many of these countries have huge areas of poverty. For kids with athletic talent sports is a great way to get out of where you are and find a better life (same can be said of many low income folks in the US as well). Baseball, like soccer is an easy sport to find equipment for (you don't kneed a glove or a real bat, just a ball and a broom handle if you really want to play). With low barriers to entry it can become a fun game for kids to play in the street and they can get quite good.
Baseball teams have capitalized on this desire to make a better life, and the abundance of athletic children with limited opportunities in their home countries and have opened academies in several Latin American countries (Puerto Rico (I know Us territory, but still), the Dominican and Venezuela are the primary ones), and have begun recruiting kids at very young ages. Children as young as 16 may be signed by MLB teams.
Until the current Collective Bargaining Agreement (CBA) MLB has been fairly lax as to how much money these players can be given (the new CBA now has a pool of money that may be spent on international signings of players under 19). Because theses players do not have to go through a draft process they are free agents and may go to the highest bidder. Recent young Cuban defectors have signed for millions of dollars upfront.
So the why, in my opinion, is a combination of lack of opportunities in their home countries, combined with a significant investment on the part of MLB teams and also the ability for teams to freely sign these players rather than force them through the draft (although this part is going away gradually).
Finally lets get some history. Obviously for the above to have happened there has to be some history here.
Baseball came to Latin America in the late 1800s starting in Cuba and Venezuela shortly thereafter. According to the Hispanic Baseball Museum Cubans have been playing in the United States since 1871. And the first Hispanic to play in the Major Leagues was Louis Castro in 1902. So we have well over 100 years of Hispanic players playing major league baseball and doing very well doing it. Significantly, black Latinos did not start playing in the Major Leagues until 1949.
According to the timeline MLB clubs have been routinely recruiting in Latin America since Cincinnati around 1911. The first baseball academy in the Dominican did not open until 1973 (although I suspect they opened in other places much earlier). There are numerous Hispanic Hall of Famers. And many more successful players that have not made it to the hall of fame. Based on this, and the fact that MLB teams continue to invest heavily in Latin America, it looks like MLB's investments in Latin America are both historic and seemingly well founded.