A phrase meaning how well something will appeal to the heartland or mainstream America. In politics, it is a gauge of how the average American will react to a policy or proposal.
Peoria is an actual city in Illinois. The phrase originated in the late 19th century, when vaudeville performers would say that if a show could be successful in Peoria, it could be successful anywhere (because Peoria was seen as an unremarkable city).
Richard Nixon popularized the phrase in political circles. He would ask how something would play in Peoria to ask how the average voter would respond to whatever he was planning.