They mean the same thing and can be used interchangably when answering a question about where you are.
Person A -"Where are you now?"
Person B-"I'm at home"
Person A-"Oh, I thought you were at the grocery store."
Adding "at" to the sentence emphasises your specific location and is slightly preferable in this situation. But "I'm home" would sound normal too.
However, when you are introducing the idea of your own location (like seeing your family after a long day at work), you would say either "I am home!" or perhaps "I am home, finally!" If it's obvious to others (your family members can see or hear you), the emphasis of the "at" is not necessary.