This is a great question. There are a few reasons why theaters in general and stars in particular gained momentum in the 19th century.
First, there was a population explosion in America. For this reason, there were more people who wanted to attend the theater. Theater quickly became the pastime for the wealthy.
Second, there was an increase in wealth. As you can imagine, when there is greater wealth, people tend to go to things like the theater. In other words, with wealth comes more time to enjoy the finer things in life. Moreover, there is also a social element to going to the theater.
Third, transportation increased during this time as well. For this reason, actors and troupes of actors could travel and perform in many different places. In other words, it was more viable to make a living as an actor. With this, actors could become very well known in multiples places. This was probably the single more important reason for the rise of "stars."
All of these elements are heightened today through technology. For this reason, we surely live in society that prizes the theater and actors.
Chapter 24 of 100 Years of American Commerce suggests that star billing dates from at least the eighteenth century.