By the late 1080's A.D., the Norman control of England was complete. Anglo-Saxon power was broken completely and the new power structure was definitively Norman. And yet, I am writing (and you are reading) in English instead of French, and in fact English is a global language while Norman French remains a regional dialect in France.
Language survival, especially in the face of conquest, is never a given thing (as the Celts ancient and modern would tell you). So how is it that Anglo-Saxon English survived to eventually become modern day English?
The following represent my own personal opinions (philologists and actual historians probably have better views):
1) Conquest Not Colonization
By the time of William the Conqueror's death in 1087 A.D., about 8,000 of his supporters - largely Norman stock, but also Breton - had settled in Britain. The guess of population in that time of English (Anglo-Saxons, Danish) was 2 to 3 million. Although the Normans and Bretons were an elite, they were definitely a minority. And William and his successors did not encourage mass migration between the two territories -after all, each inhabitant in each region represented potential taxes. Thus the Normans always remained a minority and though the Norman Language became the language of the court and law, it never sought to replace (by language or population) the native Anglo-Saxons.
2) The Plantagenet Empire
William the Conqueror, as it should be recalled, was not just the King of England. He was also the Duke of Normandy and other territories as time went on, in feudal relationship to the King of France. The Kings of England and Dukes of Normandy eventually came to build what was known as the Plantagenet Empire, with lands stretching from Ireland to the Spanish Pyrenees.
(The Plantagenet Empire - Source)
Thus, as under Cnut the Great, England was a smaller part of a larger empire. As such, the King's attention was focused not just on England, but on the domains on the other side of the channel. What mattered (where possible) was stability and tax payment, and for both of those it mattered not what language one spoke. The fact that they were ruled as separate units helped as well.
3) Trilingualism And Beyond
Early Medieval England became a sort of trilingual society with Latin, Anglo-Norman French, and Middle English being used regularly, along with other regional languages such as Danish, Welsh, Cornish, and Cumbric. Latin remained the language of the learned and the Church, Anglo-Norman French the language of the court, those that would make their way at court, and legal and financial. English remained the language of the countryside. Trilingualism is a difficult thing to manage in any age so individuals spoke the languages they needed to in order to communicate with those they needed to: likely some of the nobility knew English but just as likely most of the rural population did not know Anglo-Norman French except those that directly needed to speak with their overlords.
4) Population And Time
As noted above, Anglo-Saxon England always had a much greater population (in the thousands of percent) above Anglo-Norman French, which just continued to grow. The elite did as well; just not in the same numbers. At some point (the 15th Century) English became the majority language of the nobility and the commons, while Anglo-Norman French existed in specific realms such as the legal and diplomatic and continues to exist today in specific set statements used in the parliament of the United Kingdom.
Of course, Old English itself was not left unchanged. It had already been impacted by the influence of the Norse and Danish invaders and had begun transforming even in the 9th and 10th centuries: for example, the complicated inflectional system of nouns had begun to diminish and a new simplified definite article "the" had began to be adapted. Old English gave way to Middle English (circa 1100 - 1500 A.D.) , which in turn gave way to Modern English (1500 A.D. and following).
The last entry we have in the Anglo-Saxon Chronicle is in the year 1154 A.D. It denotes the travels of the then current monarchs, King Stephen and Queen Matilda. The last entry is not in Old English but in Middle English; fitting perhaps as the world of the Old English Anglo-Saxon had passed as well. Only someone well into their 80's or 90's at that point would remember a day when an Anglo-Saxon speaker sat on the throne of England.
Old English, like those that had used it as a living language, had faded into memory and history - yet ironically it has in large measure outlived those that overcame the culture that initiated it.
Wikipedia: Anglo-Norman French