Why was Tammany Hall involved in promoting baseball?
The term "Tammany Hall" is typically used to refer to the political machine that ran New York City for much of the time between 1850 and 1950. It existed for longer than that, but that was the time in which it was most powerful. The organization went by a number of other names as well, but it is best known as Tammany Hall.
Tammany Hall was a political machine that, for most of its time in power, used its influence among immigrants (especially Irish immigrants) to maintain its political power. Put briefly, the machine used its control of city government to provide help to immigrants. In return, it expected the immigrants to vote for its candidates. It also used its power to more or less extort money from various people--most especially those hoping to do business with or in the city.
To do all of this stuff, the machine needed to be involved in all sorts of social organizations. If it could sponsor these sorts of organizations, it would help make the people more dependent on and more loyal to the organization. Baseball clubs were one of the kinds of social organizations that Tammany Hall set up to achieve these goals. An observer of Tammany Hall once put the following words in the mouth of a Tammany Hall worker:
Another young feller gains a reputation as a baseball player in a vacant lot. I bring him into our baseball dub. That fixes him. You’ll find him workin’ for my ticket at the polls next election day.
This quote can be found in the marxists.org link.