Jane feels so guilty to having consented to a secret engagement, which was contrary to her principles and sense of right, she regarded her secret engagemnet as a "great misconduct" and felt the need to apologize to everyone whom she felt was affected -- re chap 53 --

"with the consciousness which I have of misconduct, very great misconduct, it is particularly consoling to me to know that those of my friends, whose good opinion is most worth preserving, are not disgusted to such a degree as to -- I have not time for half that I could wish to say. I long to make apologies, excuses, to urge something for myself. I feel it so very due. But, unfortunately -- in short, if your compassion does not stand my friend -- "

"Oh! you are too scrupulous, indeed you are," cried Emma, warmly, and taking her hand. "You owe me no apologies; and every body to whom you might be supposed to owe them, is so perfectly satisfied, so delighted even -- "

Jane certainly owed apologies regarding the secret engagement to Mr and Mrs Weston and Mr Churchill, as they were Frank's father, stepmother and uncle -- that is why Emma said that Jane owed her no apologies, and those to whom she owed apologies (Mr Churchill and the Westons) are satisfied to forgive her, and even delighted to accept her as Frank's fiance

Perhaps jane felt she owed Emma apologies for not preventing Frank from flirting with Emma, but according to Frank's letter, Jane had tried to stop him -- chap 50 --
"I behaved shamefully. And here I can admit, that my manners to Miss W., in being unpleasant to Miss F., were highly blamable. She disapproved them, which ought to have been enough. My plea of concealing the truth she did not think sufficient. She was displeased; I thought unreasonably so; I thought her, on a thousand occasions, unnecessarily scrupulous and cautious: I thought her even cold. But she was always right. If I had followed her judgment, and subdued my spirits to the level of what she deemed proper, I should have escaped the greatest unhappiness I have ever known".