The play opens with the great 16th-century intellectual Sir Thomas More chatting with young Richard Rich, extolling him to become a teacher rather than a politician so he won't be tempted by the bribery that comes with political office.
Soon after this conversation, More is summoned by Cardinal Wolsey, the current Chancellor of England. Wolsey wants to discuss a matter of great importance: King Henry VIII's divorce. See, Henry had petitioned the Pope some time ago to marry his current wife, Catherine, but now he wants to petition him again to divorce her. More, being a good Catholic, seems to oppose this.
Fast-forward a few months, and Wolsey is dead (after having been arrested); More is the new Chancellor. This proves to be quite stressful, however, as everyone is clamoring to know More's opinion on the divorce. At one point, King Henry even visits More in an effort to win his support. Meanwhile, Rich has been taken under the wing of one of Henry's advisors, Thomas Cromwell. Cromwell, the shadiest dude on the planet, has made it his mission to make More approve of Henry's divorce, by any means necessary.
But that doesn't work at all. After meeting with Chapuys, the Spanish ambassador, More decides to resign from his position as Chancellor. See, Henry has decided to go through with the divorce anyway, despite the Pope's disapproval, leading him to sever the Church of England from the Catholic Church. More hates this but refuses to go public about the issue so that he won't get executed for treason.
Unfortunately, Cromwell is still hunting him down. His latest plan of attack is actually quite brilliant: he convinces Parliament to create an oath declaring Henry's spiritual supremacy. More refuses to sign it but won't say why, which leads to his imprisonment in the Tower—London's most feared prison.
More thinks that he can escape execution by keeping his trap shut, but Rich betrays him by falsely claiming in court that More said treasonous things. And that's it—More is beheaded in the final moments of the play. We close things out with a character named Common Man (long story) warning the audience members to not make trouble, or they'll risk ending up like our headless hero.