"I am that I am" is a saying that means "I exist because I exist." It's like saying "I am who I am" or "I am me."
You know how you know you're you, and nobody else can be you? That's because you exist as yourself. It's the same for everyone and everything. Everything that exists, exists because it exists.
So when someone says "I am that I am," they're saying that they exist just because they do, and they are who they are because that's who they are, and nobody else can be them. It's like saying "I'm me and nobody else can be me!"