A playset was a children's toy, such as a dollhouse, which was a play area for dolls.
According to a commercial for Tribbles cereal, one such playset was the Tribble Cereal Super Salon, which included a tribble razor, which could be used to shave tribbles.
This cereal commercial was presented in a post-credit scene of the aforementioned Star Trek: Short Treks episode. The context of the commercial from the in-universe perspective is impossible to determine, and may not have been meant to be taken literally.