/u/telkanuru's answer is okay. Actually he or she is wrong at a few spots but let's flesh it out.
I'm going to be talking a lot about languages and locations in spitfire pace, so a lot of people new to the subject get lost. Here is a handy reference I wrote a long time ago.
Norman did begin to replace English, and the language of the court for roughly 200 years following the Norman Conquest was Anglo-Norman. Norman replaced and continues to be spoken on the British Isles of Guernsey, Aldernay, and Sark thanks to their proximity to France.
Over time, however, the English court began to seek to create a culture of independence - independence in terms of culture, political-economics, and (the most important) claims to the crown. This meant that the language of the people, Middle English, came in vogue. There was also a re-writing of British history to downplay the "Frankish" side to the Norman Conquest.
Anglo-Saxon speakers did not replace who they conquered. First of all, there was no extinction. There were some big fights, especially in the 400s, but the role of conquest has been over-emphasized in popular literature. More importantly were peaceful pockets of English-speaking communities where the neighboring communities dropped Common Brittonic in favor of the Anglo-Saxon dialects.
There's an easy way to demonstrate that the Brittonic speakers weren't being replaced, per se, but were changing languages. Around all of Britain are ancient Celtic counting methods that trace to Brittonic dialects that have otherwise not survived and were never written down. If you look at these different counting methods and you notice it sounds peculiarly similar to "eeny meenie miney mo" then you are not far from the truth, that is another counting method that has survived from Brittonic times. If there was an extinction or mass replacement, we would have a great difficulty explaining these regional holdovers.
Anglo, Saxon, and Jute invaders came to the Isles at a pretty ideal time. The withdrawal of the Romans left a power vacuum and the dissolution of order in much of Continental Europe allowed mass migration. Furthermore, the Germanic tribes were likely more technologically advanced than the Celtic natives. They invade with relative ease, and then are not entirely conquered from 450~ CE - 1066+ CE. That's a lot of time to conquer England. Oh, and by the way, they didn't become the dominant language of all of England. There were dozens of other tongues. What is important, however, is that the language of London, the crown, and (much later on) the university was in English.
Well, French did, a little bit. While the structure of English is fundamentally Germanic, the vast majority of English words are of Romance extraction.
The fundamental difference between the two invasions is that the first was a population migration while the second was more a replacement of the ruling class. This meant that A-S speakers replaced the people they conquered, while the Normans did not.
This latter point we can also see linguistically, with many words for 'prestige' items coming from Anglo-Norman. The typical examples are the A-S 'cu' - cow, and A-N 'boeuf' - beef, A-S 'sceap' - sheep, A-N 'mouton' - mutton; the ability to eat meat being a sign of prestige.
hi! FYI, this FAQ section won't answer your question directly, but has more background on the Anglo-Normans' transition to English