This is a funny one, culturally. Oftentimes when I walk into my office or pass someone I know at my apartment building, I will greet them with, "Good morning. How are you doing today?" Truth be told, most people who use this greeting (including myself, sad to say) are not the least bit interested in the health and/or well-being of the person they are greeting. It's just a polite way of saying hello as you walk by, and you certainly aren't expecting the other person to tell you the details of his day.
If a person responds, "I've been better," it's the last thing you want to hear. It almost obligates you (unless you don't mind being rude) to stop and continue a conversation you don't want to be involved in. Personally, I would have to ask, "Really? What happened? What's wrong?" and this could be with someone I might not even know except for passing him daily on my way to the parking lot.
Just yesterday, this happened to me while taking out the trash. My next-door neighbor, an unpleasant woman who always smells of canned tuna and cigarettes and who will complain if you give her the smallest opportunity, was coming in as I was going out. I was very careful. I did not ask, "How are you today?" I knew what would happen if I did. Instead, I simply said, "Good afternoon, Mrs. Davis."
But that didn't work either. Her response: "Is it? Not for me." This led to almost an hour of me standing in the hallway with my sack of garbage as she told me all about her lousy son-in-law, whom I've never met and don't ever care to.
Basically, "I've been better," LITERALLY means, "I'm not having a good day today" or "I'm not feeling my best today." However, you will nearly always hear it when someone means, "Oh, let me tell you all the horrible things that have happened to me today. How much time do you have?"