They mean the same actually. Generally, I am at home== formal, grammatically exact. I am home = informal,casual, daily use.

I'd use "I'm home" to announce that I have just arrived home. I'd use "I'm at home" to explain where I am at that moment.

They are both right c: But I would say I'm home not I am home For some reason it just sounds weird

I would use I am home or I'm home when announcing my arrival at home to family. Example - Mom! I'm home. What's to eat? I would use I am at my home/house if a friend called wanting to know where I was. Example - Bob - "Hey Jon. Where are you?" Me - "I'm at my house. Why?" Bob - "Some of they guys are gonna watch a movie. Wanna come?" Me - "Sure."

i am at home is formal as Aseer said but i'm home is for casual speech :)