A great question and I'll try my best to answer it. To start with a little biographical background, I always pictured him as one of the genuine rock-stars of philosophy. Described as "the keenest thinker and boldest theologian of the 12th Century" (Chambers Biographical Dictionary) he was well aware of his influence, writing in his Historia Calamitatum (A history of my Calamities, autobiography) how he gathered thousands of students. In a preface of English translation of the same book, Ralph Adams Cram wrote how Abelard was one of the most enthusiastic and daring of Mediaeval thinkers, and it is not surprising that he should have found himself at issue not only with the duller type of theologians but with his philosophical peers themselves. He was an intellectual force of the first magnitude and a master of dialectic; he was also an egotist through and through, and a man of strong passions.
It is important to mention that Abelard's teachings and writings affected him personally and professionally twice, and second incident being in the same context as Sic Et Non, that's the one we'll concentrate on. First one we'll just briefly mention: after he returned from monastery and recovered from castration (as a revenge for his affair with Heloise) Abelard started teaching theology again, concentrating mainly on the Trinity. However, a synod in Soissons was convened to examine his method of philosophical analysis. His teachings were deemed as a direct challenge to traditional ones, and Abelard was "condemned and exiled in a wild and uninhabited section of land, to devote himself to contemplation." Furthermore, he was forced to make a public avowal of faith, which he recollected later was humiliating.
Now to the main course. To understand the conflict Abelard found himself in, it's important to note that he can be observed as one of the champions of method of scholastic philosophy. As mentioned above, he was a challenger of traditional ways. He was interested in authentication of sources, in authorial intention and in historical criticism. All of which had so far been unheard of. Furthermore, he was a logician. This new critical attitude is symbolised even by the title of one of his works – Sic Et Non. For and Against. Abelard's approach was, at that time, modernist, his search for logical cogency and clarity in biblical sources was unprecedented. Sic Et Non was an assembly of 158 questions, each of which was furnished with patristic citations that imply a positive answer (sic) to the question and other patristic citations implying a negative answer (non). Abelard didn't attempt to harmonize these apparently inconsistent remarks, but in his preface he lays down rules for proper hermeneutic investigation: look for ambiguity, check the surrounding context, draw relevant distinctions, and the like.
Sic Et Non was written sometime (I like how you placed question mark in your question by the datation, as we have differently dated manuscripts of this work) after he was permitted to return and lecture in Paris, by the mid-1130s. His teachings and writings caught attention of abbot Bernard of Clairvaux. Bernard was strongest critic of Abelard’s and after unsuccessfully resolving their differences, Abelard asked the archbishop of Sens to arrange a public dispute between himself and Bernard on 3 June 1140, to settle their disagreements. He will write in his Historia Calamitatum (Chapter XII) about whole event:
"(Norbert of Premontre and Bernard of Clairvaux) ran hither and yon preaching and shamelessly slandering me in every way they could, so that in time they succeeded in drawing down on my head the scorn of many among those having authority, among both the clergy and the laity. They spread abroad such sinister reports of my faith as well as of my life that they turned even my best friends against me, and those who still retained something of their former regard for me were fain to disguise it in every possible way by reason of their fear of these two men. [...] Though I seem to compare a flea with a lion, or an ant with an elephant, in very truth my rivals persecuted me no less bitterly than the heretics of old hounded St. Athanasius."
Although Bernard accepted invitation, he secretly organized another commission with a task to review Abelard’s works on suspicion of heresy. He'll open his clash with Abelard as follows:
"We have in France a new theologian, a student of the old master, who since his childhood has dabbled in logic and has now gone mad studying the holy Writ. Doctrines which had been condemned in the past and silenced, both his and others', are brought back to life by him. Moreover, he adds new ones to them. He claims not to be ignorant of whatever is to be found in heaven above or on earth below, excepting the phrase "I do not know". He puts his mouth into the heavens and examines the divine heights, and, returning to us, he utters words inexpressible. Being prepared to use reason in order to account for everything, even for that which surpasses reason, he proudly sins against reason, and against faith. For what is more against reason than trying to go beyond reason by reason? And what is more against faith than refusing to believe what cannot be grasped by reason? For that faith in God..."
Now, the same critique he endured before first exile resurfaced. Bernard pointed out that Abelard mistakenly holds the "Arian" view that there are degrees within the Trinity. Furthermore, he attacked his Commentary on the letter to the Romans, in which Abelard had rejected the view according to which man is rightly subjected to the power of the devil from which he was to be delivered by the Son of God. Rhetorics of Abelard's sic and non especially agitated Bernard. By Bernard's words, Abelard has gone all out against the entire patristic tradition by opposing his personal non to the unanimous sic of the Fathers. For instance, sic that all the Fathers agreed on is the above mention of men being rightly subjected to the power of the devil. However, Abelard proposed non saying "as it seems to me the devil never held any right over man, except perhaps by God's permission, in which case the guard rather than the son of God should have taken on flesh in order to set man free." As Bernard passed Abelard's writings in front of the council, he condemned him for having gone transgredi terminos antiquos, quos poserunt patres nostri – beyond the ancient boundaries set by our Fathers.
For me, what's interesting the most is next part of above mentioned quote, Abelard was prepared to use reason, even for that which surpasses reason. Here we can observe proto-clash between science and religion. Abelard used logics and hermeneutics on something that, in eyes of Bernard and many others, should transcend laws of wordly reason – he attacked faith with logic. His accusers viewed his attempts as arrogant. Bernard attacked him by saying:
"All, so he (Bernard) admits, say: so it is (sic). But I say: it is not so (non sic). What do you mean? Can you do better? What greater subtleties can you come up with? Do you boast that something has been secretly revealed to you that has been overlooked by the saints and has escaped the attention of the wise?"
For Bernard, Abelard's worst sin was that he wasn't a humble servant of Gospel, like himself, but arrogant agitator. When Abelard found out that what was suposed to be a debate is de facto trial for heresy instead, he announced his intention to appeal to the Pope directly. Bernard was faster though, and Pope upheld Council's condemnation of Abelard's work. As you can see, the noise Sic Et Non produced was loud enough to reach the head of the Church. Abelard was forced to silence, and he complied by all accounts, reconciliating peacefully with Bernard. However, later Abelard will find a protector in the abbot of Cluny, Peter the Venerable, who wrote to the Pope and persuaded him to lift Abelard's sentence. He remained under Peter's wing until his death on 21 April 1142, keeping himself far from theological troubles.
Sources:
Chambers, Biographical Dictionary, 9th edition, 2011
Constance Brittain Bouchard, "Every Valley Shall Be Exalted": The Discourse of Opposites in Twelfth-Century Thought, Cornell University Press, 2003
D. E. Luscombe, The School of Peter Abelard, Cambridge University Press, 1970
Irena Backus ed. The Reception of the Church Fathers in the West: From the Carolingians to the Maurists, Brill, 1996.
Peter Abelard, Historia Calamitatum, transl. Henry Adams Bellows, Dodo Press, 2006
Peter Kin, and Andrew Arlig, "Peter Abelard", The Stanford Encyclopedia of Philosophy (Fall 2018 Edition), Edward N. Zalta (ed.)