A:  Embarrassment involves a (perceived) exposure of weakness of the self, or more to the point, of you.  When you go to a therapist, it's not some "connected part" you're examining;  you are not focusing on some part of your self--you're focusing on your self. 

An arm, for most people, is not strongly identified with -- it's just my arm, functional, but not me -- and therefore having a doctor work on it is about as shaming or embarrassing as taking your car to a mechanic. 

But with your self, the topic of therapy, there's all sorts of attachments and beliefs that get exposed to view, all manner of self-deceptions and untruthfulness that gets brought into the light, and what's mortifying, is seen by an other.  That's embarrassing.

And yet necessary.  Because, especially with anxiety and depression, our view and belief about our self is often much of the problem, and that can't get solved if that self is kept hidden.  So, oddly enough, embarrassment is required in order to heal anxiety and depression:  we have to air our "secrets of self" in order to get a new view, to really see that neither us or the world is broken.