The record player provides the musical accompaniment to the dancing referenced in the title. The man tries to draw the young couple's attention to the player several times, first offering it up for sale and then later using it to play a record so they can start dancing:
"I'm going to turn off this TV and put on a record," the man said. "This record player is going, too. Cheap. Make me an offer." (69)
At the end of the story, the young couple ends up with the record player and the young girl refers to it (and the "crappy records" (97) the older man also gave them) when telling the tale of their strange encounter with the man and his furniture.
While it's hard to tell what the older man's relationship to the record player is prior to the dancing, it seems like the record player really becomes important during the creepy, vaguely sexual dance party. So perhaps it's a symbol of whatever is going on there—you know, like frustrated or lost desire. But don't just take our word for it—what do you think?