Baseball is considered a development from an older game called rounders that was made popular in Great Britain and Ireland. It became an extremely popular sport during and just after the American Civil War. It was called "America's Pastime" because during the late 19th and early 20th century it was probably the most widely played sport in the country. Baseball was to that time period as video games and television are to today. Before TV Baseball was what kids did after school and on the weekends and during the summer...etc. It was literally how America passed the time.
Before the time of television, the internet, DVD's and the rise of American football, basketball, hockey, NASCAR, extreme sports, and the general splintering of American culture, baseball was probably one of the few things that almost every American knew something about and could discuss/share with their neighbors, coworkers, family members that they might sit down next to on a bus etc... Hence, it was the nation's pastime.
I don't think, in this day and age, we have a "national pastime" anymore. Culture is simply too splintered. American football may the most popular sport, but being the most popular sport does not make something the "national pastime." They became no life.