A kiln is a type of thermally insulated chamber or oven capable of withstanding and maintaining very high temperatures. While its specific appearance is not described in the novels, the one owned by Rubeus Hagrid was located inside his wooden cabin, Hagrid's Hut, on the grounds of Hogwarts School of Witchcraft and Wizardry. Given its function, it was likely constructed from fire-resistant materials such as brick or stone to safely contain its hazardous contents.
The primary function of Hagrid's kiln during the 1994-1995 school year was to serve as an enclosure for his newly-bred and dangerous Blast-Ended Skrewts. As the Skrewts were known to emit sparks and blast themselves forward with expulsions of fire from their rear ends, the high-temperature resistance of a kiln made it a suitable, albeit highly unorthodox, habitat for containing them within a wooden structure like Hagrid's Hut. Its use demonstrates a practical application of a mundane object for the containment of a volatile magical creature.
The kiln is mentioned exclusively in connection with Rubeus Hagrid's fourth-year Care of Magical Creatures curriculum. In the autumn of 1994, Hagrid used the kiln to house the young Blast-Ended Skrewts he was raising for his class. The existence of these creatures and their unconventional housing was discovered by journalist Rita Skeeter, likely in her illegal beetle Animagus form. She subsequently reported on it in the Daily Prophet.
The kiln plays a minor but notable role in Harry Potter and the Goblet of Fire as a key detail in Rita Skeeter's exposé on Rubeus Hagrid. During an interview with Harry Potter in a broom cupboard, Skeeter reveals her knowledge of the “shrieking, fire-breathing monsters he's keeping in a kiln in his cabin.” This information was used in her subsequent article to portray Hagrid as a dangerous and incompetent teacher, contributing to the public outcry that led to his temporary suspension from teaching. The kiln and its contents served as tangible “proof” for Skeeter's claims, highlighting both Hagrid's well-meaning recklessness and the manipulative nature of Skeeter's journalism.
There is no known additional information regarding this object from J.K. Rowling's interviews, supplementary writings, or film adaptations.