This title, along with its slightly tongue-in-cheek subtitle, do a good job of answering some key who, what, when, and where questions.
If you have any sports knowledge at all, the title "Casey at the Bat" is going to put you in a baseball frame of mind. (We know, bats are also big in the sport of cricket, too, but this poem was aimed squarely at an American "republic.")
So the title tips us that the poem is likely going to be dealing with baseball, some guy named Casey, and his turn "at bat." The subtitle sets the scene in the late nineteenth-century. At the time of publication (the same year as the subtitle date, 1888) this date would have indicated a very modern story. But for today's readers, it tells us we shouldn't be picturing today's amusement park-like stadiums. This is a more barebones, old-school baseball setting. This should also affect how we picture the uniforms and the players themselves.
The title's epigraph also identifies the poem as a ballad "of the Republic" (as in "I pledge allegiance to the flag of the United States of America and to the Republic for which it stands…"), giving things a very formal and official feel (at the outset anyway). This reference to the Republic makes a certain amount of sense. Baseball was (and in many ways still is) our national pastime.
The epigraph also suggests something epic, or heroic, is going to follow—something representing the entire country. This poem was published not so long after the end of the Civil War, when themes of patriotism and national identity were on everyone's mind (the "Battle Hymn of the Republic," in fact, was a well-known Union song). However, once we read the poem, this grandiose aspect seems a bit humorous.
After all, it's a game not a war.