Poor old Thomas More never had a chance. He rose the ranks of the British aristocracy not from copious butt-kissing or on the backs of nepotistic parents but through the sheer force of his intelligence and moral fortitude. Unfortunately for him, those same qualities become his undoing after the political order he's sworn his life to protect reveals itself to be fundamentally corrupt.
Basically, More is stuck in a big ol' pickle. Here's the deal: a few years ago, King Henry VIII specially petitioned the Pope in order to marry Catherine, the widow of his brother and the aunt of the current King of Spain. (Marrying your brother's widow was a big no-no.) Now, because Queen Catherine has been unable to produce an heir, Henry wants to petition the Pope again for permission to divorce her. The Pope isn't enthused about helping him out a second time, and More, being a good Catholic, seems to agree.
That wouldn't be a big deal, but More happens to be not merely one of the most highly respected men in England, but also the country's Chancellor. That's like being a member of the presidential cabinet. Still, even before he becomes elected to the position, More shows no desire to get involved in the debate whatsoever. Here's how he feels about the potential arrest of Cardinal Wolsey, which is the event that vaults him to the Chancellorship:
MORE: (No longer flippant) If Wolsey fell, the splash would swamp a few small boats like ours. There will be no new Chancellors while Wolsey lives. (1.380)
Spoiler: Wolsey totally croaks, and More becomes Chancellor. Although More clearly disagrees with King Henry's desire for the divorce, he even more clearly has no interest in making that private beef a public one. That's an important distinction to consider later.
Here's another important thing to remember about More: he's borderline OCD when it comes to rules and laws. It makes sense—dude's a lawyer, after all. Still, as the play continues, it becomes clear that More looks at the law with a reverence that most people save for religion and favorite TV shows:
MORE: [...] The currents and eddies of right and wrong [...] I can't navigate. [...] But in the thickets of the law, oh, there I'm a forester. I doubt there's a man alive who could follow me there. (1.690)
More has so much respect for the law because he's aware of what could happen if people stopped respecting those laws: basically, the nastier among us would run roughshod over everybody else, and nobody would be safe. The only way to prevent that, in More's mind, is to accept that the laws are what they are, and they're the way they are for a reason—unless they are changed by legitimate legal means.
Besides his love for the law, More is defined by his emphasis on staying true to your principles. We can see this in his interactions with William Roper, in which he throws shade at his soon-to-be son-in-law for wafting between fanatic Catholicism and fanatic Protestantism. We can see it in his beef with Norfolk, in which two old friends are torn asunder by Norfolk's spineless acceptance of the King's decree. Throughout all of this, More doesn't stop talking about the importance of principles. Here's an example.
MORE: [...] You see, we speak of being anchored to our principles. But if the weather turns nasty you up with an anchor and let it down where there's less wind, and the fishing's better. (1.717)
Yeah, that shade is blinding.
As you can see, Henry's little divorce scheme hits More on every front. It hits him as a devout Catholic. It hits him as a stickler for the rules. It hits him as a man who values staying true to his own principles above all else. What a lovely little trifecta of nastiness. To make things worse, More is being pressured by both sides—by Henry and his posse and by loyalists to the Catholic Church—to come clean about his opinion on the divorce, which is something that he really doesn't want to do.
That's because More, ever the lawyer, knows that he can escape execution by keeping his trap shut. Getting imprisoned for life isn't a great option—true—but it's a lot better than losing your dome. This is just about the most More-ish thing that More could do, leveraging his unparalleled knowledge of the British to run circles around Cromwell and his cronies. And it seems to work—at first. Even after Cromwell has Parliament pass an oath with the express purpose of getting More to approve of Henry's divorce (and the formation of the Church of England which allows that to happen), More holds strong and stays silent.
The only problem is that More's not dealing with honest enemies—he's dealing with corrupt politicians (isn't that a tautology?). In many ways, More's execution becomes inevitable from the moment the King asks for a divorce: there's zero chance that More will back down, and zero chance that Cromwell will admit defeat. You know—the whole "unstoppable force" versus the "unmovable object" thing. Despite this, we find this final exchange between More and his wife Alice quite telling:
MORE: Alice, you must tell me that you understand!
ALICE: I don't! [...] I don't believe this had to happen. (2.649-650)
And she's absolutely right. Thomas More could have done what every other British aristocratic did—lay down on his back—but he decides to make a stand. Key word: decides. Although it's right to look at More's execution as a tragic event, the raw truth is that any other outcome would mean that More wasn't the man he claimed to be. And we respect him a ton for that.