"Home" is also an adverb. You can say that a remark hit home, for instance. The adverb "home" in "I went home afterwards" you can think of as being opposed to the "out" in "I went out this afternoon."
Both "I am home" and "I am at home" are correct. Using the preposition is probably a little more common in British English even though nearly everybody says "I am home" even in Britain. Some British people regard dropping the preposition a little colloquial. There was a television series entitled Yes, Minister, in which a fictitious character called Sir Humphrey Appleby took great care to