This is an old vaudeville line referring to the fact that Peoria, Illinois is in the heart of the American heartland. If folks like it here, where theater was extremely popular, there's a good chance that plenty of others -- throughout the Midwest and the United States as a whole -- will like it, too.
The phrase was revived during the Nixon administration, when many of the President's senior advisers had backgrounds in marketing and advertising, a supposedly important criterion for a new policy was whether it would "play in Peoria." By that time, marketers often used Peoria as a testing ground to determine whether new products or advertising gimmicks would appeal to "average middle Americans."
The question is still used today in market research and theatrical circles with this meaning. (Really -- in recent years K-Mart field-tested a new store look and layout and Hardee's tested a new menu in the Peoria area before anywhere else in the country.) It's a cliche that Peoria newspapers bat around with proud regularity, which is interesting when you realize that it's not really much of a compliment. Sure, it's flattering to think that Peorians are considered to be so representative of Midwestern America and of America as a whole, but it also implies that we really have nothing with which to distinguish ourselves as a city.
Not that I'm complaining -- it's silly to define yourself by where you live rather than who you are and what you like. And the reduced cost of living compared to the suburbs of Chicago or St. Louis makes a little blandness entirely worth my while... but that's another node.