Literally, a hinge is the metal joint that lets a door swing. Take the hinges off and the door is not held to anything any more — it falls, it swings wild, it does not shut properly. That picture is the whole word.

As a verb, to unhinge someone is to shake their mind so badly that they stop thinking or behaving normally: 'The loss of his wife unhinged him.' It is a strong, rather formal, rather old-fashioned verb, and you will meet it far more often in books than in speech. Note that it is almost always something big and painful doing it to a person — grief, fear, months alone. You would not say a small annoyance unhinged you unless you were joking.

The past-participle form 'unhinged' is where the word really lives today, as an adjective. Older use: mentally ill, not in control of one's mind — 'an unhinged man wandering the streets.' Said this way it is serious, and quite cold; it is not a kind word for a person who is genuinely sick.

But here is where you have to watch the age of the speaker. Since about 2018, younger people online use 'unhinged' for anything wild, out of control, with no filter — and they usually mean it as a compliment. 'That ad is completely unhinged.' 'Her posts are unhinged and I love them.' A brand posting nonsense at 2am, a friend sending forty messages in a row, a film with no rules — all unhinged, all praise. So one word carries a heavy clinical meaning in a novel and a cheerful 'this is gloriously silly' meaning in a group chat, and only the situation tells you which. If you are not sure of your audience, the older meaning is the one people may hear, so use it carefully about a real person.