I'm an English teacher, but our training tends to be pretty vague on the events of the fifth and sixth centuries around the time Anglo Saxon/Old English appears. I studied some of this in graduate school in the 1990s, and, at the time, the narrative presented was that Anglo Saxon culture eclipsed Celtic culture as the result of large-scale invasions of Germanic tribes. I know from the History of the English Language course I took that very few Celtic words were incorporated into English.

At the same time, some recent British archaeologists using ground-scanning techniques are saying that they are not finding the evidence they thought they should find of large-scale invasions and are hypothesizing that the Anglo-Saxon takeover was accomplished by a relatively small number of invaders. (They are also arguing that writers like Bede and Gildas may have been making things up in their accounts.) The whole topic seems to be muddied by a lot of sentimental attachment to mythical histories.

It may simply be a failure of my imagination, but I have a hard time imagining how Anglo-Saxon language and culture would have replaced Brittonic Celtic so completely without large scale invasion, especially since there are so many examples of native languages persisting in the face of colonization. The Irish never entirely lost Irish Gaelic. English persisted in modified form after 1066. Indians speak English, but also Hindi and Urdu. Are there counter-examples of languages being abandoned in favor of the language of new arrivals?

I understand that the documentation of the period is poor at best, so the only answer may be that we don't yet know exactly what happened.