Hello! You asked about the supernatural in The Castle of Otranto.
If you are a fan of gothic literature, you might be interested to know that Horace Walpole's 18th Century novel is the precursor for all future Gothic fiction. Walpole influenced a slew of Gothic authors, including Bram Stoker, Edgar Allen Poe, and Daphne du Maurier.
In his novel, Walpole attempts to combine Old Romance with New Romance. Old Romance greatly focuses on the supernatural and the fantastic. New Romance is more down to earth. In combining the two, Walpole is able to present ordinary characters working amidst extraordinary circumstances. 18th Century England saw the reign of Henry VIII, who started the Anglican Church in order to circumvent Catholic Church rules about marriage. The king wanted to marry Anne Boleyn but the Church was not keen on dissolving his first marriage to the Catholic queen, Catherine of Aragon. All Gothic architecture at the time was almost always religious in structure: they were churches, cathedrals or monasteries. Henry VIII had many of these Gothic (Catholic) churches dismantled or turned over to the state. In due time, the destruction of a religious heritage, the fascination with the unknown, and the religious struggle between Catholic and Protestant England came to embody Gothic literature. The persecuted woman of Gothic fiction paralleled the persecution of the Catholic Church by Henry VIII.
Walpole's Gothic novel is filled with fantastic occurrences: whether it is a giant helmet falling out of nowhere and crushing a lord's son to death, or the ghost of Manfred's grandfather stepping out of a portrait, or a gigantic foot suddenly appearing out of thin air and another gigantic hand resting on the bannister. Walpole seems to want to tap into his readers' fascination with the world of the unknown and the terrifying. The question of inheritances, of successors and of marriage is heightened through Walpole's sometimes surrealistically supernatural elements.
Remember that the 18th Century ushered in the Age of Reason. Science, the art of deductive logic and observation were all openly lauded by Enlightenment thinkers. However, the everyday person did not cease to be fascinated and enthralled with the exotic world of magic and superstition. Walpole's familiar and sometimes farcical supernatural elements allowed his readers to relate to his story. They allowed his readers room to contemplate their own changing world: Manfred's fight to marry Isabella and secure his heir mirrors Henry VIII's bold move to create a new Church, exempt from the power of the Catholic Church.
Hope this helps you understand the place of the supernatural in Walpole's entertaining novel. Thanks for the question.