How do I remove this brown staining on the back of this plate?

You could try hydrogen peroxide.

Here's an article about what to do, and more important, what not to try! I know I have a few older pieces of pottery/plates with this very same staining. I haven't messed with trying to remove the stains as I figured you can't really see and I just figured its some sort of chemical type change under the glaze. But if you have luck, let us know and I might attempt it as well!

http://www.lakesidepottery.com/Pages/Pottery-tips/removing-dark-crazing-lines-and-stains-from-porcelain-and-pottery.htm

Try soaking the dish in milk whole milk for about a hour then rinse

The info at this link may help: http://www.lakesidepottery.com/Pages/Pottery-tips/removing-dark-crazing-lines-and-stains-from-porcelain-and-pottery.htm

I get tea/coffee stains inside my glass containers and fill then with very hot water and add about 2 tablespoons of oxiclean and soak. They come right out!

I had stains on my Corelle cups. I tried bleach but then I used a damp magic eraser and the stains were gone.

I use the magic eraser on stains with no problems.

Mr. Clean Magic Eraser works! I use it all the time on my tea cups and also the coarse areas on the bottoms of dinnerware that gets dark stains.