Some context first. In the XVIIIth century, the Church had enormous political influence in France. Almost the entire population was catholic and, most importantly, the majority (around 80%) of it was living in rural areas. For this part of the population, the local priest was extremely important, and not only for spiritual matters; it was generally him who provided what education the peasants' children received, it was generally around him that rural dwellers who tended to live relatively isolated gathered every Sunday. He was a key component of rural society and could influence the inhabitants of his parish in political matters as well as spiritual ones, though this could vary depending on the region. In the region around Paris or in Provence, the degree of attachment to the Church was generally lesser than in Brittany for example, though keep in mind that in the end it always came to a personal opinion, which means that no generalisation can be made with too much certainty.

When the Revolution broke out, the desire to reform the Church was strong, for several reasons. First, its members, the clergymen, were among the privileged ones who were living off taxes and obligations imposed on the peasants, notably the tithe; however it was far from the only one, as the Church as a whole (meaning abbeys, bishoprics) possessed around 8% of the kingdom's lands, and in these land lived people who could be taxed by clergymen in other ways, sometimes in kind (pieces of harvest, unpaid labor, etc).

There was also widespread criticism of the luxury in which the high clergy lived, notably from the lower clergy, a good portion of which often lived in conditions similar to that of the peasants. Other internal disputes focused on theological or even political matters; one particularly important doctrine that was on the rise in this period was gallicanism, the idea that France should have its own national church, though even the proponents of gallicanism rarely envisioned a full break away from Rome and remained attached to catholicism. Of course, the Revolution and the burning nationalism it brought gave new momentum to gallicanism. Another reason pushing the Constituant Assembly to reform the Church was the somewhat common idea that the Church was spreading obscurantism and superstition in the population, and that as such it was in its present state incompatible with the new enlightened world the Revolutionaries were creating. This feeling was confirmed in many minds by the high proportion of clergymen (like the Abbé Maury, one of the most notable anti-revolution members of the Assembly) within the most reactionary part of the assembly.

It was thus decided to write a constitution for the clergy : it would not bring any significant changes to the rites or to theology, however it would bring major political upheaval, as it declared the abolition of regular clergy (monastic orders were dissolved) and clergymen to be employees of the state, who would receive a salary from it, and swear an oath to the national constitution. At first it can seem like not much, but the key part was the oath thing. For many clergymen, the only oath they could take was the one they had with God; pledging another would be betrayal. Furthermore, rejecting the oath was a way of manifesting one's opposition to the Revolution more generally, and as Pope Pius the sixth began to condemn the civil constitution and the Revolution in general, a full blown schism progressively developed, with around half of the clergy taking the oath (that's the constitutional clergy), the other half refused (that's the refractory clergy). Those who belonged to the latter were stripped of their offices and replaced with priests who belonged to the former. This created enormous tensions all across France, as millions of people suddenly saw the priest of their parish fired, with someone else, sometimes from a different region, standing in his place. It is key to note that each territory saw a different proportion of priests accepting or refusing the Constitution.Here is a map that depicts these proportions : the brighter it is, the less numerous the were to take the oaths. As you can see, it was acceptable for some; for others, it was not, especially when the refractory priests were popular, or when they began to denounce theirs colleagues as "false priests", or even traitors.

Combined with the other issues that were haunting the country (like the inequal way in which the lands confiscated to the Church had been redistributed through auction sale and in some places largely acquired by rich urban dwellers rather than by peasants, or the fact that several feudal taxes and obligations were in fact still legally implemented), the religious rift created enormous tensions all across France. Counterevolutionaries were outraged by what they perceived as an attack on the Saint Church, while revolutionaries were angered by the refusal of the refracory priests to consider nationhood to be as important as God, often suspecting them of spreading counterrevolutionary ideas. For the former, those who had refused the oath were righteous men who remained loyal to God, the Church and generally the King as well, while those who had taken the oath were dubbed "juror priests" and deemed as heretics, especially since they supported a Revolution which had granted equal rights to Jewish and protestant communities of France, something woefully unacceptable for the most conservative. For the revolutionaries, those who took the oath were good priests, dedicated to freedom and the nation, while the refractory clergy was seen with increasing suspicion, and eventually as a hatred bunch of traitors working hand to hand with the tyrants of Europe (currently at war against France) and manipulating the peasants into revolting to overthrow the Republic. In early 1792, the legislative assembly voted to incarcerate them, a decision which was promptly veto-ed by the King : the religious schism was now a national division that opposed the King to the Assembly. The former lost popularity as a result and was finally overthrown in the summer. From then on, nothing could stop the government from attacking the refractory clergy, which in turn fully embraced counter-Revolution.

This was nowhere clearer than in Vendée : there, the clergy had overwhelmingly rejected the oath, and had thus been ousted in favor of unpopular "juror priests"; the inhabitants of the nearby cities like Nantes had been almost alone in benefitting from the auction sale of confiscated Church lands and were way more supportive of the Revolution than the countryside, who often tormented their new priests in more or less violent ways. When full blown civil war broke out in 1793, refractory priests were always accompanying the rebels, who inevitably demanded freedom of worshipping with them and no one else when negotiations took place with the republican forces. Lazare Hoche, a republican general who was appointed as de facto dictator of the western regions to deal with the rebellion in 1795, managed to discourage the rebels by granting them such freedom if they renounced to fight, among other things. The Civil Constitution of Clergy was without a doubt one of the most (if not the most) divisive issues of the entire Revolution.

Sources :

Timothy Tackett, The Coming of the Terror in the French Revolution (2015)

Jean-Clément Martin, Nouvelle Historie de la Révolution Française (2012)