To be anxious means to have anxiety/discomfort.
The subjunctive is followed by certain verbs and to desire is one of them. So when a subjunctive follows anxious it follows a similar pattern and the meaning is "desire strongly to the point of causing one anxiety, discomfort."
Of course the anxiety/discomfort may not actually exist and the expression is used for exaggeration or emphasis.
So it does in a sense combine the meanings of both anxious and desire when used like this.
I'm anxious that you meet my friends.
This sounds fine and the speaker/writer is saying he/she wants you to meet his/her friends very very much.
But, you can also say something like this:
I'm anxious that they will not make it here in time.
This means the possibility of "they will not make it here in time" is making you anxious.
I'm anxious that he hurt himself.
It's unlikely that a reasonable person would be anxious to hurt someone, so probably most people would assume that hurt here is past tense and that you are saying that the possibility of him having hurt himself is making you anxious.
I'm anxious for him to hurt himself.
if you want to express without ambiguity that you just really wish someone would hurt themselves.