In British English, you say "I'm home" when you have just arrived home.
For instance, you might call someone, after a journey, to say that you've arrived home safely:
1. I just called to say I'm home!
You use "at home" for any other situation:
2. I'm at home today.
3. Is there anyone at home?
4. "Where's your husband?" "Oh, I left him at home."
I don't know whether it's different in American English, though.