Why was Joan of Arc burned at the stake?

Joan of Arc was formally burned for being a heretic. At her trial, she was arraigned on three indictments. The first was that she was a witch who had used magic. This was related to her confession that she had heard the voices of St. Michael, St. Margaret, and St. Catherine telling her to dress as a boy and fight against the English in The Hundred Years' War. The first indictment, like the other two, expressed the prevailing wisdom concerning the role of women in society. Women were not supposed to fight; this was purely man's work. By heading off to battle, Joan was undermining the established social order. Her trial can partly be seen, then, as a way of restoring that order.
The second indictment was that she was headstrong in speaking for the faith. This is related to the first indictment in that Joan was doing something that females of the time weren't supposed to do. They were expected to be faithful and pious, but in a quiet, demure way. Joan was anything but quiet or demure.
The third indictment was based upon Joan's claims to receive inspiration directly from God. Before the Reformation, this was considered highly dangerous, not least because it undermined the position of the Church as intermediary between God and man. In claiming to receive her orders directly from God, Joan was attacking the Church's spiritual authority. According to her accusers, this was little more than rank heresy bordering on outright blasphemy.
 
https://www.history.co.uk/this-day-in-history/30-may/joan-of-arc-is-burned-at-the-stake