Short answers, pulled from the story.
The Ecclesiastical Appeals Act 1532 was a law passed by the Parliament of England on the 2nd of April 1533 that declared England an empire and the monarch an emperor. This legislation forbade all appeals to the Pope in Rome on religious or other matters, making the King the final legal authority within England, Wales, and other English possessions.
King Henry VIII needed the Ecclesiastical Appeals Act 1532 to annul his marriage to Queen Catherine of Aragon because the act removed the Pope's ability to intervene or uphold the marriage. This legal change allowed Thomas Cranmer to grant the King his divorce so he could marry Anne Boleyn and secure a male heir for the Tudor dynasty.
Thomas More, the former Lord Chancellor, was executed for his refusal to acknowledge the King as the supreme head of the church following the Ecclesiastical Appeals Act 1532. His execution occurred because he refused to acknowledge the annulment of his parents' marriage and the King's supremacy, which was treated as high treason.
Princess Mary wrote a letter to King Henry VIII on the 22nd of June 1536 acknowledging the annulment of her parents' marriage. This public submission saved her life but marked the end of her status as a legitimate heir under the new legal framework.
The whole act was repealed by section 1 of, and part II of the schedule to, the Statute Law (Repeals) Act 1969. Specific sections were repealed earlier by the Statute Law Revision Act 1950, the Criminal Law Act 1967, and the Ecclesiastical Jurisdiction Measure 1963.