It's a bit rubbish really. The Jesuit professor in charge of the case against Bruno (Robert Bellarmine) drew up a list of accusations, but this doesn't exist anymore so we have to work on subsidiary information found in Bruno's works and Bellarmine's own notes, rather than the actual trial documents.

Angelo Mercati is the man who found and published the documents surrounding the trial and subsequent condemnation, and in his view, Bruno's crimes were completely of a religious nature - his view of the cosmos isn't important to the trial proceedings from a scientific perspective because it's not science - it's religious, as Bruno denies the virginity of Mary, the divinity of Christ and a number of other heterodox positions. It was religious in Bruno's eyes too, and therefore the Catholic church felt justified because he was holding a heterodox position. To classify it as 'science' is to anachronistically reclassify something which it wasn't. Here's a quote from Bruno about the sun:

The cause of such [motion of the earth] is the renewal and rebirth of this body, which cannot last forever under the same disposition. Just as things which cannot last forever through the species (speaking in common terms) endure through the species, substances which cannot perpetuate themselves under the same countenance do so by changing their configuration.

Spot the great amount of scientific thought going on there.

Nicholas of Cusa and William Vorilong both argue for the plurality of worlds well before Bruno, so it's hard to see that as a problem, and Cusa was made a cardinal after he wrote about this in De Docta Ignorantia (1440). Bruno does the same, but then starts giving souls to the stars, meteors, planets and the universe - a much more radical theological departure, and probably the root of his heresy charge in this matter.

Came here from /r/Catholicism, but I keep hearing all this stuff about Bruno. I was curious, so I dug in, and concluded that he was right only in the sense of a stopped clock being right twice per day.

So far as I can tell, Bruno was an astrologer, not an astronomer, one who spent much of his time writing treatises on "magic" (incl. De Magia (On Magic), Theses de Magia (Theses on Magic), De Vinculis in Genere (Of Bonding in General), and De Magia Mathematica (Of Mathematical Magic)). He believed demons caused disease and had a penchant for worshiping Egyptian gods. Not saying that burning people at the stake was a good thing, but IMO what Bruno was doing was pseudoscience, focusing on magic and the occult, and I've never understood why self-proclaimed skeptics have such a thing for him. To be honest, from what little of him I've read, I have no idea how he came up with the idea of an infinite universe. It certainly wasn't based on scientific observation. Have I missed something?

De Grassi Tyson seems pretty resistant to modern takes (aka non-hagiographies) on the history of science in general. See http://blogs.scientificamerican.com/cross-check/2014/03/07/neil-degrasse-tyson-please-speak-out-about-militarization-of-science/ and http://www.trueanomalies.com/want-to-know-more-about-giordano-bruno/ for more discussion of how some people in the history of science community have been trying to catch him up, with little success to date.

In a lot of ways, he seems interested in science cheerleading most of all, which is a disservice in a lot of ways. Build something up as perfect and otherworldy, and the first human correction or error makes the whole thing crumble. Accept from the start that science is a social, human enterprise - and that doesn't make it 'better' or 'worse,' it just is what it is - and you won't have quite so many people thinking climate change is all a conspiracy because actual human scientists have come to conclusions by means other than absolute and unwavering rationality and apolitical meritocracy.