"I'm home" or "I'm at home"

@Mitch I don't think you're completely correct. Some people say something like "I'm home, I'll talk to you later" over the phone so in that situation you may be announcing your arrival or coming home to someone who is not present in your house. "I'm at home right now" is really only a response used when initiated by a question as to where you are; when you say "I'm home" it refers to your unexpected arrival and is typically only used either to end or begin a conversation depending on the way in which you use it.

I think the implication in the expression "I'm home" is that you're home from somewhere. It may, as Mitch says, be that you've just come/gone in, but it doesn't need to be — you can be home from the front or home from university and have been back for a week or so. But a homecoming in the not-too-distant past is certainly connoted.

The nature of the word home in "come/go home" is often argued over; noun and adverb are suggestions, but directional particle is what I'm voting for at the moment. Apparently, home in go home can be used without a preposition because it is the remnant of an earlier form that fused preposition (to) and noun, and when this type of inflection for case disappeared from English, this particular usage continued, with the to-home form standardising with the base noun. We don't have "I'm going school". Possibly, "He is home" has arrived via "He has/is come to-home". Home in "He is home" is locative rather than directional now.

It may be another legacy from the fused to-home that we never use the preposition to with home (except with an intervening determiner etc.), but this is true for other nouns also.

Certainly, other prepositions (at, from) can be used with the noun home in the expected way, and "I'm at home" is strictly locative. To express arrivals at other venues, we are forced to use a different expression — "I've just got to school / I've arrived at school", as psmears says.

"I'm home" means "I have arrived at home" (after being away for while, for instance after a trip, or after working in an office the whole day).

In the phrase "I'm home", the word "home" is used as an adverb, not a noun. Compare the expression "to go home".

Example: in the movie Star Wars: The Force Awakens, when Han Solo and Chewbacca set foot on the Millennium Falcon for the first time in many years, Han Solo says "Chewie, we're home". (YouTube link)

Another example: in a TV sitcom set in the 1950s, a husband might return from work and loudly announce to his wife "Hi honey, I'm home". This would only make sense if, for instance, she is in the kitchen and might not hear him entering the house, so he has to announce his arrival. It would not make sense if she is already at the door to greet him.

Note however: sometimes the expression "X is home" is used as an abbreviated version of "X is at home". For instance, "We rang the doorbell, but nobody's home" = "Nobody is at home", no one answered the door.