Gary is correct. "I am fine" is often just a polite response. Maybe I'm sick, maybe I'm not. For instance, I may be as sick as a dog, but if someone asks me "How are you?", sometimes I may just reply with "I'm fine" as I don't wish to talk about how I really feel. So, a response of "I'm fine" is often just a polite conversational response, rather than an indication of how you really feel.

I think it depends how you say it. Depending on the tone, it can mean you are fine, or that you are not fine, but are trying to either be polite, or that you don't want to say how you really are.

The others make good points, but one thing not noted is that "I'm fine" is not what you would say if you wanted to describe a very good condition and a high degree of willingness to engage with the other person. That is, it's sort of neutral and somewhat discourages more talk. For example, it's the stereotypical thing that a teenager would say to a prying parent, or an angry spouse might say to the other if he wasn't in the mood to talk. So it can be polite and meaningless, but it's definitely not as positive/encouraging "I'm great" "Things are going really well" "I'm getting used to being retired/a sophomore/a new parent."

Gary and Greg have both explained it well. "I'm fine" is a polite response, but it's often fairly meaningless, since people say it all the time, whether or not they are really "fine." As Gary says, the speaker's tone is more important than the words. Does the person sound cheerful? excited? depressed? irritated? The usage doesn't have anything to do with being American, by the way. (I'm American, but Gary is from New Zealand, and Greg is from Australia. It seems we all use this expression the same way.)

"I am fine" is perfectly correct to say. However, depending on the conversation and the tone it is spoken with, it could be a passive-aggressive way to mean that you are not okay.