The answer boils down to a matter of whether the Catholic Church should retain the power to dictate what is "truth" or not.
The Catholic Church was used to have a monopoly on the interpretation of the Bible, and since God was the ultimate authority and truth, in practice the church had the monopoly on all kinds of truths in Western Europe.
This did not in itself mean that there was no intellectual discourse, it was just that it was held within the church. Before the renaissance, if you were an intellectual, you were generally a monk or a priest. As such Cardinal Nicolas of Cusa published radical cosmologies that removed Earth from the centre of the universe, and made it a star, like other stars (remember that the planets then was thought to be stars). He did this in the 15th century completely without being threatened by burning.
You could get executed as a heretic at this time even of you were clergy, of course, but to do so you generally would have to either subscribe to a theological opinion that had been deemed a heresy, or otherwise challenge church authority.
But in the early 16th century things changed with the rise of protestantism, which denied the Pope and the Catholic Church’s ultimate authority to decide which interpretation of the Bible was true.
At the same time the Renaissance meant that many intellectuals now were not monks or priests, and therefore philosophical/scientific claims were increasingly being made by people who were not a part of the church. This also challenged the church's monopoly on truth.
In 1543 Copernicus book placing the sun in the centre of the universe was published. Copernicus himself avoided problems by dying before his book was published, but the book sparked discussion within the church. Although Giordano Bruno held several heretical views and was burned at the stake it's unclear if his cosmological views had anything to do with that. He seems to have based himself more on Cusa than on Copernicus, and as such his cosmology was not a direct challenge on the church authority. But it may have contributed to the discussion about cosmology withing the church, a discussion that ended the 24th of February 1616 when a commission of theologians decided that the heliocentric view "explicitly contradicts in many places the sense of Holy Scripture" and hence was a heresy.
As a result of that decision Galileo Galilei was asked to stop teaching Heliocentrism, and when he didn't he was put to trial, where he recanted his views completely to avoid execution.
In the centre of the commisions decision of 1616 is the question of what should be the ultimate truth, reality or the church. And the Catholic Church unsurprisingly decided that it should remain the church. The unstated reason for this is that they were worried that accepting a heliocentric view would make the whole theology crumble.
And although the catholic church has only slowly accepted the otherwise more common consensus, that science gets to look for earthly truths and the church takes care of the heavenly, I can't find any references to the church actually threatening to execute scientists after Galileo. They did continue to add scientific books to the list of forbidden books though.
In the end no-one was ever burned for being a scientist, but the Catholic Church attitude to science and the trials of Galileo did put the Catholic Church on a collision course with science that didn't really end until John Paul II, who admitted that Evolution was not just a hypothesis and Vindicated Galileo.