The next piece of the phrase is what it means when I say, "I am angry at you." What "at you" means is "all of you," not part of you or your actions. To a child, this phrase can feel very intimidating and overpowering. The child tends to interpret the phrase as them being "bad or wrong," not what they did. The thought to consider is that we are powerless to change who we are, but we do have the power to change our behaviors and choices.
What we are probably meaning to say when we communicate emotion is, "I feel anger (frustration, confusion, irritation) with what you did." This phrase takes a few more words to state, but I hope you can appreciate the accuracy of what is being stated. The most accurate way to communicate emotions is with the verb "to feel," not the verb "to be."
Another issue to address is that when we state our emotion with the verb "to feel," we can "feel" more than one emotion at a time, but it is difficult to "be" more than one emotion at a time. The second part of the communication, "at what you did," addresses the issue of the action of the individual, not the individual himself. When we address the action or behavior of the person, it is very clear what they can change.
In educating children and adults on the "semantics of emotion," I often tell children that their parents will always love them, but they may not like what they do. If we address the action, behaviors and choices of the individual ("at what you did"), it points out what they can change. If we address the individual as the object of our anger ("at you"), it is difficult to know what to change.