How do I thank a professor for choosing me as his PhD student (via email)? I emailed my potential supervisor to express my interest in her works. We talked once and I was encouraged to apply to the university. Recently, I emailed her to inform her that I have submitted my application. I got a response from my potential supervisor that I am admitted. How should I thank her for admitting me? 

Also, one of my recommendation letters is not submitted yet. Does that mean I am definitely admitted? Because I have to ask about financial aid and scholarships and I have no idea when or how to do so...
How should I thank her for admitting me?


Just like you would phrase any other similar email, within or outside of academia. There is no need to overthink this. There is no specific protocol for that, and no need for a special "thank you" email (although an answer that acknowledges that you have received the mail and are still interested in the position is definitely advisable).

Dear XXX,

I am very happy to hear this, and I am looking forward to working with you.

Can you please advise me on what the next steps for me are?

All the best,
Mahsa Gamiji

As a quick sidenote to all those expressing disbelief that a single professor may have decided to accept or reject a PhD candidate: This is how it works in most places in Europe. Don't assume that admission to PhD school is organized the same as in the US everywhere.