Where It All Goes Down
No question about it, "Casey at the Bat" is set at the—you guessed it—ballpark. But remember, this poem was written in 1888 and ballparks and ballplayers looked quite a bit different than they do today. If you're picturing this, you're off base. Nope, this is more along the lines of what Thayer had in mind.
The name of the town, "Mudville," also affects how we view the poem's setting. It sounds small-town-ish. We are more likely to picture this ballpark in a rural setting than an urban one. We might imagine that Mudville's hometown ballpark might even be a little run down. (It's the name that makes us feel that way. Sorry, Mudville.)
Importantly, this small-town, baseball setting also reinforces the poem's bigger setting (as does its epigraph): America. When you hear that something is American as baseball and apple pie, that's no accident. This poem goes a long way toward painting the sport as the country's quintessential pasttime, setting it right smack dab in the middle of the American heartland.