How do I talk to my professor about my anxiety/depression?
First, everyone feels overwhelmed in grad school, and nearly everyone suffers from imposter syndrome.  It isn't just you who feels like everyone around you is better.  This isn't likely to help with your illness, obviously, but I think every grad student is better off once they internalize that idea.

As @ff524 pointed out in the comments, the first thing you need to do is go talk to your school's disabilities office.  You don't need to go in with a clear picture of how it goes or what you want out of it - you just need to go in and explain the situation to them just as you did to us.  It's their job to help you, and very often that includes facilitating conversations with your professors as well.  

You don't have to go to your professor and throw yourself on their mercy.  I understand why you would be traumatized from the awful experience you described, but understand that what that professor did wasn't just wrong, it was likely illegal. Faculty have clear obligations that the disabilities office will help them understand and work through, just like they help you.

You should also get in touch with a mental health professional.  If you're in a new town away from your old one, go to your school's health center and I'm certain they can refer you.

Basically the only "wrong" thing you could do is to continue doing nothing.  Get in and get help right away, and don't feel even remotely bad about doing so.  This is exactly what those resources are there for.