Let me give you a summary based on three memorable points.
Context. First, you need to know that the play takes place in the 1530s. During this time, most of Europe was religious. In other words, religion played a bigger role than it does today, at least in the West. Within this context, King Henry VIII wanted to have children. This was important, because England was a hereditary monarchy. So, the bearing of children was part of the reason why kings married. However, Catherine, his wife, could not have children. So, he wanted a divorce.
Conflict. Here is the main conflict. The Catholic Church would not allow divorces. Cardinal Wolsey attempted to organize a divorce but failed. The Catholic Church stood its ground. When Wolsey was dismissed, Thomas More came into power. The play is named after him, I should add.
This is where Thomas Cromwell, the antagonist, suggests to the king to leave the Catholic Church. The king does. The church of England is formed. But the King needs Thomas Moore to sign the divorce papers. But because Moore is a pious Catholic, he cannot. Furthermore, he is a man of integrity.
Conclusion. Because of Moore's integrity, he is imprisoned and later beheaded under Cromwell. Moore, therefore, is an example of virtue and sacrifice, which is the point of the play.
I can see your confusion, as it is a historical play. In other words, you need to know something about the Reformation and the Church of England to understand what is going on. Here is the context:
The play is set in the 1530s and the events of the play are rooted in history. The protagonist is Sir Thomas More, who is devoted to the Catholic Church.
King Henry VIII (who would have six wives) realized that his Queen, Catherine, would not be able to bear a child. This was a huge issue during this time, because in a hereditary monarchy, an offspring was important. So, he calls upon Cardinal Wolsey to organize a divorce, which was not allowed in the Catholic Church. Wolsey fails, as the laws of the Church trump Wolsey's attempt. This is where Thomas More comes into office.
But there is a complication here. Thomas Cromwell, the villain of the play, advises the King to break away from the Church and start a new one. King Henry agrees, succeeds, and creates the Church of England and obtains a divorce.
Thomas More, because he is a man of principles and integrity, has to go against the divorce, as his religious scruples would be offended. He therefore does not sign papers to make the divorce official. So, he is imprisoned and later Cromwell sentences him to death. He dies as a hero and martyr (he is beheaded).
The point is about integrity and moral courage. This quote sums it up:
Well . . . I believe, when statesmen forsake their own private conscience for the sake of their public duties . . . they lead their country by a short route to chaos.