It might seem like a very ambitious title since this is a complex question but there are some simple reasons that are at the heart of the longer and more complicated answers.
On the 22nd of May, 1430 John of Luxembourg lay siege to Compiegne. He captured Joan and held her for ransom. In November of that year she was handed over to Pierre Cauchon, Bishop of Beauvais, to stand trial as a heretic. One May 28 of the following year, she was condemned as a heretic and sentenced to die. On May 30, 1431, at the age of 18 or 19 Joan was burned at the stake. She cried out to her Lord, "Jesus, Jesus!" until she could no longer speak. A Parisian who kept a diary wrote that she was soon dead and the coals were racked back to show everyone that she was a woman. She was given no dignity in death. Then the fires were started back up and her body burned to ashes. The ashes were thrown into the Seine.
One of the first reasons is the ambition and greed of the Bishop of Beauvais, who sat as a judge at her trial and got her from John of Luxembourg who was holding Joan for ransom. His advancement was tied to the English claim to the throne. Another is that like Margery, Joan would not submit to the authority of the church. She insisted that she heard voices and that they were miraculous and she insisted that she had the right to wear men's clothing as she was wearing them to protect her chastity.
But, one has to wonder why, since she had done so much for the Dauphin Charles, that he did not ransom her even though he had the means and the opportunity? One of the conditions for ransom was that Joan cease to fight against the English, something that she would not promise to do. Another problem was that, when she attacked Compiegne, she did not do so with Charles' permission. She was acting on her own, with her own men, not the Dauphin's army and she attacked Paris on the feast day of the virgin for which she was also condemned.
Still she did Charles a great service.
The legitimacy of Charles VII was a concern since his father Charles VI was mentally incompetent and his mother Isabella of Bavaria had been unfaithful, but Henry VI's claim to the throne was no better since he was related to the French crown through the female line and under Salic law could not inherit. Also Isabella had been unfaithful with her brother-in-law, Louis of Orleans, and if he was not Charles VI's son, he was still in line for the throne. However, at the time of Joan involvement with the Valois cause, they had lost heart and the Dauphin was unsure of himself.
What Joan did at Chinon in 1429 to convince Charles to fight the English and that his claim to the throne was supported by God is unclear. At her trial, her account changed at first she pleaded that she had sworn not to tell but, under pressure and probably torture, she spoke. Her voices lead her to pick out Charles in the crowd even though she had not seen him before and she revealed to him a prayer that he had secretly made and told no one about. Also, the voices are supposed to have shown him a sign upon Joan's appearance that Charles VI was his father. She said that the voices wanted him to raise the siege at Orleans. This military victory accomplished gave the Valois cause new heart.
One of the things that is said is that Joan was not a virgin. This is only being put about by the English side to justify her murder since, if she was a virgin, her conviction as a heretic was problematic since the devil does not deal with virgins. After she was captured and brought to Paris, she was examined by the Queen of Sicily and her ladies and found to be intact. Even some of the English captors said she was a virgin, possibly to protect themselves from charges that she had been sexually assaulted in custody.
For whatever reason that she was killed, it was a terrible to die and it failed as Charles VI entered Paris four years later. By 1450, the English ended their struggle for France. I recommend Marina Warner's book Joan of Arc, The Image of Female Heroism for those who want to read more.
