This is a nice question. I'll respond a bit to some of the particulars you are asking about, but nothing actually beats sitting down and reading the book yourself, honestly.

So, the context: Patricia Crone and Michael Cook were students at the School of Oriental and African Studies, and both spent time interacting with the great John Wansbrough, himself an academic titan. Wansbrough was not just a scholar of the Islamic world, however, but someone well-trained in biblical criticism, and his own body of work was looking to re-evaluate the origins of Islam and Qur'anic scripture. He was skeptical of early Islamic sources and their ability to seemingly resist internal criticism.

Finishing their PhDs in the mid 1970s, Crone and Cook set out to write Hagarism almost immediately, heavily inspired not just by Wansbrough, but by their own concerns over the problematic nature of the Islamic sources more generally. What were these problems? First, if we are hoping to reconstruct "what really happened" during the rise of Islam and the establishment of the earliest caliphates in the seventh/eighth centuries CE, we lack almost any contemporary Islamic evidence which describes these events (what we do have is limited to things like brief inscriptions and numismatic material, for instance), and are instead forced to rely on what tend to be called the "narrative" Arabic sources which date from, at best, the ninth century CE in written form. There was little research done at the time concerning the reliability of oral/aural transmission of traditions in the early Islamic world in the early period. We also have to address the hagiographical and romantic aspects of these texts which, of course, purport to describe the events contemporary to their prophet and the exempla which gave birth to the religion and realm these later authors now served. When Crone and Cook were writing in the late 1970s, too, they also lacked sources which would come to light in later decades (see, for instance, something like the surviving portion of Sayf b. 'Umar's Kitab al-Ridda), let alone the change in approach that their work itself would cause/inspire.

What was that long lasting change? The two, in many ways, heralded the true arrival of what Fred Donner would later dub the "skeptical approach" (Narratives of Islamic Origins, 1998) to early Islamic sources, although they weren't the first to begin seriously questioning the available source material. Their prospect was bold: the early Arabic/Islamic material is so problematic as to be irreconcilable. And so, they turned instead to available contemporary and near-contemporary evidence they could find from the non-Muslim perspective, and came to many of the bold (and now almost completely discounted) suggestions that you find in Hagarism, Wansbrough's Qur'anic Studies, and Crone's later Meccan Trade and the Rise of Islam. Crone's attitude to the problems of the Islamic sources is best summed-up by her magisterial introductory chapter to her 1980 book Slaves on Horses: The Origins of the Islamic Polity, which is born out in an oft-quoted line:

one can take the picture presented [in the early Islamic sources] or one can leave it, but one cannot work with it.

To sum up in brief: many argued that what they did was to over-correct concerning the problems of the Islamic sources, really. To say that nothing could be gained from these narrative sources aside from reading Muslim narrative is something that many of us (although not all of us!) would say was foolish, but they ignited a debate that still rages today. Can we rely on the things these later sources are telling us? Can we really use them to tell us what was actually happening, rather than to twist any historical fact and reflect it through a later lens contemporary to the author? Even though the book has been heavily discredited and Cook has addressed many times since its issue the problems of this early work (Crone, for her part, never did before her recent death, BTW), for anyone writing about the earliest centuries of Islamic history, you must engage in some form with this vital dialogue started by the two - even if just to explain why you disagree, and why you are doing something different.

Then, of course, is the huge amount of later research their work inspired - far too much to list here! The most obvious, in many ways, being Robert Hoyland's work which was born from his PhD dissertation (under Crone!), Seeing Islam as Others Saw It, which nuanced his teacher's previous contributions and brought together a much larger sampling of non-Muslim writings contemporary with the period.

To say that the work of Crone and Cook has been momentous in the field of Islamic history and studies wouldn't quite say enough, I would argue. Even if they were wrong in their results when they started.

I'll leave you with Chase Robinson's wonderful discussion of Crone's life and work which appeared in Crone's festschrift.