From the first known execution of heretics in western Europe in the middle ages - when 16 people were burned at Orléans in 1022 - burning at the stake became an emblem of the righteous orthodoxy, righteous justice. In the next two centuries about 150 heretics were burned - either by secular or Church judgement, it's hard to see clear responsibility until the late 12th century when the Church formally took on clear jurisdiction of heresy.
From the late 12th century onwards the burnings of heretics surge, in particular in the south of France during the wars of the Albigensian Crusade where hundreds are found to be burned at stroke. For example at the siege of Minerve in 1210, according to the Cistercian monk and witness Peter of les Vaux-de-Cernay:
Our Count [Simon de Montfort] now entered Minerve and went to the place where the heretics were gathered. As a true Catholic he wished them all to win salvation and come to know the way of the truth, and he began to urge them to be converted to the Catholic faith. When he could make no progress, he had them taken out of the castrum. There were at least 140 perfected heretics. A huge pyre was made ready, and all were thrown on it - indeed there was no need for our soldiers to throw them on, since they were so hardened in their wickedness that they rushed into the fire of their own accord. ^1
This progressed to more formal organization of burning-as-punishment with the formal authorization of the medieval inquisition in the 1230's under Pope Gregory IX, where judgement would be rendered by the Church and fitting execution by burning handled by the secular authorities.
As the investigation of heresy broadened in the 13th and 14th centuries, through such inquisitors as Bernardo Gui and Nicholas Eymerich, sorcery began to be suspected as a heresy although there are few convictions for it until the late middle ages. In fact the Catholic Church never reconciled itself to the existence of witchcraft per se, and rarely sent anyone to execution for it. However, secular authorities showed no such resistance, generally because witchcraft, or maleficium, was usually an issue not of religious crime, but of murder or crime against property. Witchcraft prosecution - notwithstanding the theories of demonologists who came from the Church and from within Protestant communities as of the 16th century - was almost entirely a secular prosecution. However, the punishment maintained its heritage as heresy: burning at the stake.
Brian Levack in his book The Witch-Hunt in Early Modern Europe tells us that while convicted witches were burned in Europe, they were not always alive when torched:
Most witches were not burned alive. Although this was the usual practice in Spanish and Italian territories (where relatively few witches were executed), witches in France, Germany, Switzerland and Scotland were usually strangled or garroted (a brutal procedure in which a metal spike penetrated the throat) at the stake before the flames consumed their bodies. Many French witches were hanged before their bodies were burned. In the German principality of Ellwangen the princely prebend usually changed the sentence of burning to one of death by the sword, but he still had the corpse burned afterwards. In similar fashion an entire group of witches in Ortenau were sentenced to death ‘by the sword and burning’ in 1630. In Sweden it was standard practice to have condemned witches first beheaded and then their bodies burned. Occasionally other methods of execution, such as drowning, were employed, but in the overwhelming majority of capital sentences either the witch or her corpse was burned. ^2
So why were heretics and witches burned, treated to different execution than other secular crimes? Heresy was a crime against God. Medieval clerics, in particular monks, developed broad allegories of maintaining God's garden based on New Testament scripture which is full of allusions to pruning and purity, like Christ's words in John, chapter 15:
“I am the true vine, and my Father is the gardener. He cuts off every branch in me that bears no fruit, while every branch that does bear fruit he prunes so that it will be even more fruitful. You are already clean because of the word I have spoken to you. Remain in me, as I also remain in you. No branch can bear fruit by itself; it must remain in the vine. Neither can you bear fruit unless you remain in me.
The infected, the rotten, the unredeemable, are clipped and treated like garden rubbish:
“I am the vine; you are the branches. If you remain in me and I in you, you will bear much fruit; apart from me you can do nothing. If you do not remain in me, you are like a branch that is thrown away and withers; such branches are picked up, thrown into the fire and burned".
^1 Peter of les Vaux-de-Cernay (tr. Sibly & Sibly), The History of the Albigensian Crusade (Boydell, 1998), p 85.
^2 Brian Levack, The Witch-Hunt in Early Modern Europe (Pearson, 1987, 2006), p 94.