Well, part of the problem was that he thought most of the Church officials were idiots (or "asses" in his own words) and he had low tolerance for stupidity. However, more interestingly, he was fascinated with large numbers and he believed in an infinite cosmos with multitudes of solar systems. Very cool! Some of these ideas, of course, also led to problems with the orthodox Catholic beliefs. From the New Yorker review:
The bronze figure of Giordano Bruno that stands at the center of Rome's Campo de' Fiori may be the most successful commemorative monument in the world. The average statue in a park or square usually rates no more than a glance: Either you already know who the guy is, or you don't care. But the hooded and manacled effigy of Bruno, with its haunted stare, immediately catches the eye, and the gruesome story attached to it -- Bruno was burned at the stake in that very spot, for the crime of heresy -- cements him in memory. Practically every tourist who comes to Rome tromps through the Campo and hears that story, even if they've never heard of Bruno before. The students who commissioned the statue in the 1880s, as an emblem for freedom of thought and the division of church from state, really got their money's worth.
But who was Giordano Bruno, and why was he executed in the Campo de' Fiori in 1600? A common misperception mixes him up with Galileo, who ran into trouble with the church 16 years later for embracing the Copernican model of the solar system instead of endorsing the Aristotelian belief that the sun revolves around the Earth. (In fact, the two men shared an Inquisitor, the implacable Cardinal Robert Bellarmine, canonized by the Catholic Church in 1930.) Bruno, too, thought that the Earth circled the sun, and subscribed to many other than heterodox ideas as well: that the universe is infinite and that everything in it is made up of tiny particles (i.e., atoms), and that it is immeasurably old. But as Ingrid Rowland demonstrates in her new biography of the renegade thinker, "Giordano Bruno: Philosopher/Heretic," Bruno was no martyr for science. What got him killed was a murky mixture of spiritual transgression and personal foibles, combined with a large dose of bad luck.
If there were countless worlds besides ours, this sidelined the Christian story. Creation, expulsion, salvation: such things might have happened, but somewhere off in a corner, while other things were happening on other planets. Also eliminated was God’s difference from humanity. If, as Bruno saw it, God was present in every atom of the universe, then transubstantiation became a silly idea. (God was already in the wine.) Ditto incarnation. Bruno later said that he started having doubts about Jesus at the age of eighteen; in his mature philosophy, the Messiah has no place. Nor does original sin, or pretty much any sin. God “makes his sun rise over good and bad,” Bruno wrote. Even devils were going to be pardoned. To lead a virtuous life, you had only to do unto others as you would have them do unto you. As the reader may have noticed by now, much of this constitutes liberal Christian thought in our time. (What Bruno discarded was the Church’s literalism—exactly what many of today’s believers have done.) Likewise, Bruno’s cosmology anticipated modern physics and astronomy. But it did not accord with the views of the sixteenth-century Church. It sounded like Protestantism, or worse.Ultimately, there was a combination of factors, including his personality, that did him in (from Salon):
It was what Rowland calls Bruno's "combative personality" that finally did him in. The Roman Inquisition, in an especially insecure and punitive mood on account of widespread Protestant agitation against the church, had only the Venetian nobleman's testimony against the philosopher. Then one of Bruno's former cellmates, a man he'd slapped during a dispute and who feared that Bruno had informed on him as well, stepped forward to relate the various blasphemies and heretical convictions Bruno had spouted during their time together behind bars.
The last straw was Bruno's refusal to accept the authority of the Inquisition itself. Even so, his rebellion was peculiarly Catholic: He kept insisting he'd recant if the pope personally confirmed to him that his beliefs were heresy. This infuriated Cardinal Bellarmine, known for his conviction that harsh punishments make good teachers. Sixteen years later, Galileo managed to elude the more extreme penalties meted out by Bellarmine and company with a public (and essentially politic) repudiation of his heliocentric views; he lived to fight another day under a relatively comfortable house arrest. Bruno was characteristically less prudent, and died naked and gagged (by some accounts with an iron spike through his tongue), in flames.
As Rowland points out, Bruno, irascible as he was, had committed no crime, not even the disruption of mass, a common practice by militant Protestants of the day (and also punishable by death). He "had done nothing in his life except talk, write and argue." When his fate was pronounced, he told his condemners, "You may be more afraid to bring that sentence against me than I am to accept it." It took a long time for that to prove true, yet thanks to those idealistic 19th-century students, everyone who comes to Rome to behold the splendor of the Vatican is also presented with a reminder of its bloody, repressive past. Every year, on the anniversary of his death, free-thinking Romans cover his statue with flowers. While the church has since expressed "profound regret" for his persecution (which it simultaneously tries to palm off on "civil authority"), this can't be comfortably reconciled with the canonization of Bellarmine a mere seven decades ago. Dead 400 years and largely unread but immortalized nevertheless in bronze, Giordano Bruno is still a thorn in their side.
Read the reviews from Salon and the New Yorker. Both of these reviews are excellent. The New Yorker review is longer and provides more details about his philosophical ideas and his work in the arts of memory (this also played a role in his downfall).