'So am I' means 'I am xxx, too'. For example:
'I'm thirsty'.
'So am I. Let's go and get a drink.'
'So I am' is a less usual phrase, but we sometimes use it mean 'You are right! I am xxx.'
'You're covered in pencil shavings!'
'Oh, so I am! I didn't notice. The lid must have come of my pencil sharpener. How annoying. I'll just go to the wastepaper bin and shake them all off. Thanks for telling me!'
Be careful how you use these expressions, because they aren't interchangeable.