To be afraid is to be scared. People are afraid of snakes, bugs, heights, bears, death, and public speaking, among other things.
Being afraid means you have a fear of something — and people are afraid of all sorts of things, from guns to ghosts to being in arguments. People also say, "I'm afraid I have some bad news," which doesn't mean they're afraid of the news, but sorry that they have to share it. It's a similar situation for the saying "I'm afraid so," which is a nice way of saying "I'm sorry, but the answer is yes."