If your mother accuses you of stealing her necklace and pawning it, you'll answer her indignantly that you did no such thing!
Indignantly is the adverb form of the adjective indignant. If you are indignant about something, you're offended and mad because someone did something unfair. You might indignantly demand that you are paid equally to your co-worker who does the same job. If you get an F on a paper you worked hard on and that you know is good, you might indignantly demand that your teacher explains the poor grade.