All emphasis mine. And there are a couple sides to this: the secular authority logic, the religious angle, and Papal confirmation of the tradition.

The origin of the practice may be in Diocletian's persecution of the Manichean sect, where it was commanded that they be burnt alongside their scriptures, on the Rescript to the Manichees (302). It was viewed as harsh punishment, and different from usual religious persecution, which hadn't commanded the burning of scriptures. This is mostly a personal opinion; if any Roman experts want to refute it, I'll edit it out.

Now, for the part where I do have more documentation.

On the psychological effect of a death such as burning in those onlooking.

From De Heretico Comburendo Act - 1401

(...) and they the same persons and every of them, after such sentence promulgate shall receive, and them before the people in an high place cause to be burnt, that such punishment may strike fear into the minds of others, whereby,no such wicked doctrine and heretical and erroneous opinions, nor their authors and fautors, in the said realm and dominions, against the Catholic faith, Christian law, and determination of the holy church, which God prohibit, be sustained or in any way suffered...

This is a secular law, and it clearly implicates burning as a coercive strategy, not only a punishment grounded in moral reason. Public burnings of heretics were relatively commonplace, and had an 'exemplary' value.

We can refer to the Summa Theologica too, for Saint Thomas view on it.

Question 11. On Heresy - Article 3. Whether heretics ought to be tolerated?

(...) Wherefore if forgers of money and other evil-doers are forthwith condemned to death by the secular authority, much more reason is there for heretics, as soon as they are convicted of heresy, to be not only excommunicated but even put to death. On the part of the Church, however, there is mercy which looks to the conversion of the wanderer, wherefore she condemns not at once, but "after the first and second admonition," as the Apostle directs: after that, if he is yet stubborn, the Church no longer hoping for his conversion, looks to the salvation of others, by excommunicating him and separating him from the Church, and furthermore delivers him to the secular tribunal to be exterminated thereby from the world by death.

The emphasis is on repentance. The long process of burning allowed for recantation until the last moment. During Giordano Bruno's execution, litanies were sung, and he was constantly petitioned to recant, so that their soul may be saved from Hell as his body died.

On Papal authorization, we know - from Exsurge Domine, papal encyclical condemning Luther's teachings - that the Catholic Church considers the burning of heretics a correct teaching:

(...) In virtue of our pastoral office committed to us by the divine favor we can under no circumstances tolerate or overlook any longer the pernicious poison of the above errors without disgrace to the Christian religion and injury to orthodox faith. Some of these errors we have decided to include in the present document; their substance is as follows: (...) 33. That heretics be burned is against the will of the Spirit.

I hope this helped at least a bit!