First, I think Eliot invokes “That Shakespearean Rag” to demonstrate the degeneration of culture from Shakespeare to the modern era. The context of where the Rag intrusion appears is a clue: “A Game of Chess” opens with a stanza of ancient grandiosity, and becomes more base as the part progresses into the modern era, where the Rag comes in. It’s as if Eliot is saying, “So this is what we’ve come to?” To him, all of the great ancient thoughts and feelings– like the works of Shakespeare– have become modern and cheapened in popular culture.
Second, Eliot includes the Rag to show the disorienting nature of the modern world. The song appears completely out of nowhere, disrupting a conversation between two speakers. If only he had lived to experience ringtones! Newer technology was beginning to intrude on the lives of people in unwanted ways. The song’s sudden inclusion, then, also shows what it’s like to live in a hectic fast-paced world. Add the song’s disruptive quality with the low culture aspect, and you get an obnoxious aberration of past culture that represents the worst that modernity has to offer. To Eliot, of course– I kind of like it.