Wha happened when Anne got up to recite at the concert? How did Anne count herself ''rich''?

When Anne stood up to recite, she experienced an acute case of stage fright.

Anne was the victim of an overwhelming attack of stage fright.

In answer to the second part of your question.....

"We ARE rich," said Anne staunchly. "Why, we have sixteen years to our credit, and we're happy as queens, and we've all got imaginations, more or less. Look at that sea, girls--all silver and shadow and vision of things not seen. We couldn't enjoy its loveliness any more if we had millions of dollars and ropes of diamonds. You wouldn't change into any of those women if you could. Would you want to be that white-lace girl and wear a sour look all your life, as if you'd been born turning up your nose at the world? Or the pink lady, kind and nice as she is, so stout and short that you'd really no figure at all? Or even Mrs. Evans, with that sad, sad look in her eyes? She must have been dreadfully unhappy sometime to have such a look. You KNOW you wouldn't, Jane Andrews!"