There's no definitive answer to this question - there's not a whole lot of evidence, hence the change in scholarly opinion. You're right in assuming that the local Brittonic population was larger than that of the Anglo-Saxons - there were, perhaps, 1 million Britons in Britain during the 5th century and far, far fewer Anglo-Saxons.

It's probably useful looking at the end result of the Anglo-Saxon settlement. By the 7th century, Anglo-Saxon kingdoms had formed in the most heavily Romanised zones of Roman Britain, occupying the eastern and southern parts of the island. The kingdoms were so thoroughly pagan that they needed a papal intervention to convert to Christianity. A Germanic, Anglo-Saxon identity, distinct from the local Welsh, had also emerged - Ine's laws, issued at the end of the 7th century, have clear provisions for Welsh and English people. Old English presumably had emerged as the dominant language as well. There's very little shared between Welsh and English - indeed, Italian has had a greater influence on the modern English lexicon than Welsh. This presents the most extreme situation in the barbarian west - in the rest of the Roman west, the barbarian kingdoms were thoroughly influenced by the native Roman culture.

Bryan Ward-Perkins gives two reasons why Anglo-Saxons did not adopt more from their Welsh neighbours. Firstly, it took an exceptionally long time for the Anglo-Saxons to assert themselves militarily over the Brittonic peoples. Elmet, a Brittonic kingdom in the eastern part of the country, persisted into the seventh century, and western Britain continued to be Brittonic for a long time. Secondly, Roman control and ways of life had broken down in ways that were unparalleled elsewhere in the Roman west. Towns and Roman life ceased to be a thing by the start of the 5th century. There may have been a drop in Latin literacy, too. Political authority also seems to have fractured. Nonetheless, there is evidence that the local Britons still saw themselves as "Romans", in a sense. In contrast, a still somewhat Roman Gaul was conquered quickly by the Franks. It may be so that the invading Anglo-Saxons did not see Brittonic culture as something worth emulating - both because they were constantly in contact with those people practicing their culture and there was not some sort of imperial past that the local rulers could aspire too. It has been suggested elsewhere that Anglo-Saxon paganism was perhaps a strong identity marker that differentiated them from the local Britons.

It therefore seems to be the case that the Britons within the Anglo-Saxon kingdoms in the east, which were militarily and politically dominant, adopted an Anglo-Saxon identity, along with their language and religion. Burial practices during the migration period change, with "continental" practices introduced. It's impossible to know the ethnicity of those buried and it's highly likely that those who may have considered themselves Brittonic in the past also chose to be buried in Anglo-Saxon ways. The dynasty of Wessex seems to have had some sort of Celtic origin, too - some of the names of the early kings of Wessex, such as Cerdic, Ceawlin and Cynegils look like they have Brittonic origins. That subjects would adopt the culture of their superiors is not uncommon - in Hungary, the local Slavs would become Magyarised; across the Elbe, Slavs would become German; and in Pictland, the local Picts would adopt the Gaelic culture of the dominant Gaels of Dal Riada. One can see this process happening in Devon, which despite being part of Dumnonia and conquered quite a bit later by Wessex, would adopt English place-names and English culture. This elite emulation model would explain the disappearance of Brittonic languages.

Slightly more controversial is Alex Woolf's argument. [EDIT - clarified his argument below, I garbled it here - there's both his interpretation and alternative interpretations in the following]. The aforementioned lawcode of Ine gives different blood-prices - weregelds - for British and English people. If I remember correctly, the British values are considered around half that of the English - it's certainly significantly less than their English equivalents for similar social ranks. It would appear that the main characteristic differentiating the two groups was language. Woolf argues that these lower weregelds were part of a deliberate policy to gradually reduce the status of the Welsh by disenfranchising them and transferring their wealth to the English. There was, at the very least, a strong incentive to identify as Anglo-Saxon. There's no evidence of similar strategies being employed elsewhere, but it is a possibility that similar laws existed in other Anglo-Saxon kingdoms that created a strong financial incentive to identify as Anglo-Saxon - there unfortunately just isn't that much evidence for this period on the whole.

In short - perhaps buoyed by financial incentives, the Brittonic speakers appear to have emulated the culture of their overlords. The overlords, meanwhile, were not keen on adopting local cultural elements, perhaps due to a sense of discrimination and constant conflict between them and the native population. Unfortunately, due to the lack of evidence, this is largely speculative - we can never know the exact reasons why so many Britons adopted the culture of their invaders.

Sorry for the long response, hopefully that answered your question!

Sources:

Ward-Perkins, ‘Why did the Anglo-Saxons not become more British?’, English Historical Review 115 (2000)

Lucy, The Anglo-Saxon Way of Death (Sutton, 2000)

Woolf, A., ‘Apartheid and economics in Anglo-Saxon England’, in Britons in Anglo-Saxon England, ed. Higham, N. J. (Woodbridge, 2007)

This question's a few days old but I just wanted to offer an alternate perspective since I study historical linguistics.

First, popular history tends to paint the pre-Saxon Britons as a homogeneously Celtic-speaking population. However, there's a growing body of work from a number of linguists (for example, Peter Schrijver) who argue that British Latin was far more widely spoken than has been realized, and was possibly the majority language in the southeastern part of the island. So in the initial areas the Anglo-Saxons settled, it's entirely possibly that Brittonic (or Brythonic, but it seems that for whatever reason scholars prefer Brittonic) was rarely heard. Arguably the strongest piece of evidence for this--and the notion that there was a large withdrawal of these Romano-Britons from eastern Britain--is the Brittonic spoken in western areas of the island, which in the fifth and sixth centuries seems to have undergone a large and swift morphological simplification of the sort that tends to accompany language contact. There are also a large number of Latin loan words in Welsh. So the change in Brittonic at this time was been exponentially greater than any change that occurred in Old English before the Viking period, and this change seems to have been the result of contact with Latin rather than English.

So what all this implies is that (a) Latin was probably fairly widely-spoken in southeastern Britain and (b) many of these Latin-speakers seem to have moved west, possibly as a result of the migration of Germanic-speakers, which means that if around 200,000 speakers of proto Old English moved into areas like East Anglia and Lincolnshire, they would have become numerically and culturally dominant quite easily. That they did is borne out by early "-ham" place names (typically given in the fifth century) such as "Walsham" (homestead of the Welshman) and "Brettenham" (homestead of the Briton) which reveal where Britons remained but also that they were rare enough that their ethnic identity was a distinctive feature, enough to be used as a toponym.

It is true that if we take a figure of 200,000 migrants, the original population likely outnumbered them by something like five to one. However, demographic shift is not an uncommon phenomenon, particularly when one group is reduced by a series of plagues which arrived through Roman trade links, which is exactly what seems to have occurred with the Britons. The Anglo-Saxons in the east of the island, with their North Sea trade links, were not so badly affected. Additionally, many Britons seem to have decided to move to Armorica at around this time, creating Brittany. All of this meant that it would have been fairly easy for the Anglo-Saxons to expand westward into the somewhat depopulated Brittonic-speaking regions, and as they did so, they probably intermarried with the locals, creating genetically and culturally mixed but linguistically English communities (this can be seen in the Celtic names of some early West Saxon and Mercian kings such as Cerdic, Caedwalla and Penda). The dominance of English was probably decided by the fact that the emerging Anglo-Saxon kingdoms were part of a North Sea world, in close contact with other Germanic speaking regions. In these regions, English was likely the main spoken language by around 650 CE.

In the final areas to become English-speaking prior to the Norman conquest (such as the Welsh marches and Cumbria), it is likely that little more than elite dominance would have been necessary to affect a language shift. But the speed of the transition in other parts of the country, as well as the sudden changes in western Brittonic, certainly suggests that a large-scale change in the makeup of the population occurred. In absence of such migration, language shift is a long, drawn-out, uneven process that takes centuries. For example, it took six hundred years for Latin to eradicate Gaulish even with all the might of the Roman Empire. More modest elite groups such as the Normans, the Franks, and the Visigoths never managed it at all. That a small band of Anglo-Saxons could have for some reason accomplished it in all of England in less than the time it had taken the Romans to convert only southeastern Britain to Latin certainly stretches the boundaries of believability.

At the time Bryan Ward-Perkins wrote the essay referred to in the other answer, the popular interpretation of the Anglo-Saxon migration (amongst archaeologists at least) was indeed that it was little more than a warrior elite that acculturated the natives. Archaeologists have since turned in a more "migrationist" direction again, particularly now that DNA evidence from ancient graves has actually shown that whole communities moved, rather than just men, and that in many cases the migrants seemed to have been less wealthy than the natives. After all, one of the major reasons for their migration seems to have been that their marshy homelands were being regularly flooded.

If you want more information and sources, feel free to ask.