How do you tell your friend that you really miss them and want to be friends with them again?

Don't beg them. It depends how you fell out. If your old friend now has a lot of new friends and spends all their time with them and tends to make fun of other don't bother asking to become friends with them because chances are she'll start making fun of you.

I suggest that you start talking talk her a little bit everyday but don't keep talking about things you did together. Maybe if you share a class talk to her simply as classmates then talk to your other friends out of class. Eventually you'll become friends again.