In those times there was no modern political theory, kings depended on the Church for legitimacy.
If you did not believe in the Church, you probably do not believe in the King either. God and Country were intrinsicaly tied.
So the Kings were very jealous to stamp out heresy. And in those times Churches took salvation very seriously. False doctrines were sending people to Hell and had to be stopped. Is like those annoying evangelicals or JW of today that what to save you and keep pestering you because they are concerned for your salvation but their methods were much more serious them. That set the pace for the inquisitions.
There were three.
The Papal inquisition in Italy that was the most benign.
The French inquisition to stap out the Cathari heresy and the infamious Spanish inquisition.
But remember, torture was a normal procedure in those times not only in the Inquisition but in most court of law, also burning at the sake (death by asfixiation) was more bening that quartering or dismembering, normal ways to execute people in those times. The Inquisition was more benign that most civilan courts of the times.
The Papal inquisition was from the Pope.
The French and Spanish ones were actually established by the Kings, not by the Church, king that knew if heresy spreads in France they will loose their crowns.
The Spanish have a huge problem with large numbers of Jews and Moslems in Spain that did not believe in the divine rule of Queen Isabella.
Even the number of deaths was relatively small.
The popular Catholic/Protestant withhunts of the 17 century killed more people, as the war of religion at that time.