The period of time in question is about 80 years after the Norman Conquest, William the Conqueror was succeeded by his son William Rufus. Who in turn was succeeded by his brother Henry I. Henry had only two legitimate children (and about 20 illegitimate ones) - William and Matilda. William died young, drowned when the White Ship sank in the English Channel in 1120, and so Henry had no male heir. He promptly re-married but that marriage had no children. So he reluctantly designated Matilda as his heir, and made his nobles swear an oath to support her as heir.
Matilda had been married to the Holy Roman Emperor, she was sent to Germany at the age of 8 and educated there. When she married at the age of 12 she started to take on the role of Empress - in Germany the wife of the ruler was to some degree a co-ruler, so she granted charters etc. Wilkinson was saying that when Matilda was 12 her exercise of power was probably under the guidance of the Emperor's advisers as part of her education, part of her training to rule. Once the Emperor died in 1125 she was summoned to return to England by her father to be designated as his heir, and re-married to Geoffrey d'Anjou in the hopes that this second marriage would produce offspring (it did).
Despite saying that she was his heir it seems that Henry didn't really do much to make sure she had a chance of holding the throne. All three experts were in agreement that he didn't let her establish a power base of any sort in England - he assigned her no lands, no castles. So really he was responsible for what happened after he died in 1135 - instead of Matilda inheriting, the throne was seized by her cousin Stephen de Blois. And the nobles in England were all perfectly happy to let this happen. The nobles in Normandy would've preferred his older brother to take the throne, but no-one was really on Matilda's side (or at least not publicly). This was partly because she was a woman, and partly because she was foreign-educated. She was also widely regarded as proud and arrogant - Wilkinson was clear that she thought this was primarily because Matilda was a woman. That the same characteristics and actions as a man would've brought Matilda praise. The other two seemed to think that there was more truth to this than that - that Matilda might've been able to make life easier for herself if she'd been a little less concerned with her status as Empress.
Stephen was the sort of man who got along with everyone - he'd not been intended to be King & in many ways stayed more first-among-equals with the Barons, rather than their ruler. Carpenter was fairly anti-Stephen, he thought that he showed poor judgement in choosing who to please, whose side to take in disputes. Gillingham felt rather that Stephen had inherited a bad situation, and did as well as he could. But whichever is true, after a while Robert Earl of Gloucester (one of Henry's illegitimate children) went over to Matilda's side. He escorted her to England and his holdings gave her the power base she'd not had before. Stephen had the chance to capture Matilda at one point during this journey, but didn't do so - which Gillingham thought was the right course of action due to the potential effects on Stephen's reputation, but Carpenter thought was a ludicrous mistake.
The conflict dragged out for nearly 20 years, although there weren't many actual battles. Stephen's wife, another Matilda, was instrumental in both the negotiations and in raising armies particularly during a period where Stephen had been captured by the Empress. She wasn't regarded with as much distaste by the nobles, because she managed to do this while still behaving femininely enough for the standards of the time. Despite the lack of battles the war had a lot of effect on the country - hence the later name of the Anarchy. One of the standard strategies in warfare at the time was to ravage the lands around your opponents castles - so burn the crops, burn the villages, ruin the economy of the area as well as deny the fortresses food. Gillingham and Carpenter disagreed on how much and how widespread this was. Carpenter was presenting a picture of the whole country in flames and turmoil, but Gillingham felt that outside a few areas it was pretty much business as usual for the peasantry.
The war was finally over when Stephen and Matilda's son Henry came to an agreement that once Stephen died then Henry would be heir.