This is a one of those questions that is particularly interesting because it's quite hard to actually answer. Contrary to the popular belief espoused by Gildas and Procopius (and later Bede) that the Angles, Saxons and Jutes killed or displaced the native Britons, archaeological evidence suggests a much more piecemeal process of settlement which often overlapped or actually intermingled with pre-existing British communities. Indeed, some Early English polities actively identified themselves with pre-existing British kingdoms, and some retained identifiablty British elements until at least the 9th Century. The English and the Britons overlapped and mixed their communities and their material culture, so why not their language?

Some British elements did enter English: Kent, for example, is derived from the pre-Roman Cantwara people who inhabited the region; Eoforwic, the English name from which we get York, is likely a bastardisation of the Brittonic Eburākon or Eboracum, the Roman name for the site, and just consider the number of River Afons in England. Of course, the vast majority of the functional lexicon of Early Medieval England appears to have been in Old English. This might be because English was an elite language. Consider that British society in the 440s and 450s was one that had been devastated by an incredibly virulent plague, famine and invasion; the English in contrast seem to have been vigorous and enthusiastic newcomers. In communities where they integrated, such as Mucking, Essex, they appear to have frequently been able to work more tenuous land around settlements and greatly expanded agricultural production. If an English presence served to increase trade and connection between communities, it is perhaps only natural that Old English become a lingua franca in those contexts.

Ok so this is complicated and can be broken into 3 seperate but related questions. 1) Who were the Britons? 2) What language did they speak before invasion? And 3) Why did they relinquish their language?

So 1) Britons in this day and age this is a loaded question, what does it mean to be British? It can mean anything from being part of the "British Isles" thereby including Ireland. It can mean from the island of Great Britain thereby including Wales and Scotland. Let's assume you mean specifically the area of England. Now within England 3 main Germanic tribes conquered 3 different regions of Britain. The tribes were the Angles, Saxons, and Jutes. There was no "Anglo-Saxon" at this point they were seperate and distinct. Now the areas settled ranged from Kent to the eastern coast of England and not too far inland. History evolves to tell us that it wasn't until the middle of the Middle Aged that England was unified but up until then there were 5 main kingdoms. Three of these begin now with the Germanic invaders. Now of course people lived in these areas already we'll call these the Britons. These Britons as evidenced in Roman accounts were organized in a tribal basis in a traditionally Celtic system (Think Boudicca and the Iccene). These tribes were not and never had been unified! That's why the Romans were able to conquer the areas so relatively swiftly. The Roman presence though quickly fell apart and Britain fell out of the empire. This left the Romanized Britons and the rest of the citizens on the island to hash things out. But it's important to note the Britons were never very thoroughly Romanized. In areas such as London and Bath sure they had a presence but it was never enough to alter the average Briton's life in significant ways. Most Britons kept living as they had. Their religion was also mostly intact as well as their language.

2. So what language did they speak?? Well we don't know. What we do know that Welsh and Irish borrowed heavily from two other languages of a Celtic origin. At the branch split between P-Celtic this is evident. In the North we assume the larger borrowing of the two to be the now lost Pictish and the south in Wales we assume it to be Brittonic. Notice I leave out Scottish Gaelic (it arrived as a borrowing of Irish in the early millennium from Iona and Colum Cille). But there's an important language missing! It isn't Manx or any others in the British Isles. Actually if we look to France in the region of Brittany we find out answers! Brythonic is the language of Brittany and is used even to this day (look up singers such as Nolwenn Leroy). If we examine Brythonic it seems to fill the gaps of the loan words in Welsh and Irish! This tells us that the language of Brittany is likely very similar to the old language of Brittonic.

3. Alright now we can address your question more directly, why not assimilate from the Germanic languages to Brittonic? Well why would they? The natives when not killed were unlikely to want to remain in a place the Germanics were taking over so those that could fled. They fled to Brittany, Wales, and wherever they could. And those that stayed? They assimilated. They were never well organized as they had a tribal structure, and in previously Roman areas Latin while great was never the lingua franca it was elsewhere in the empire. So these loosely affiliated people never had a singular language to unite them. There was no nationalistic tie to their language as there is today. They simply needed to be able to communicate with not just each other but one another. It's like asking why English is the language of use in the USA and not Navajo or Apache. The locals needed to be able to communicate and since the Germanics had one language and the locals had many it was easier to just use the Germanic language to speed things along. At least it was initially. Overtime loan words would creep in and blend the two languages together. More, in the later Medieval period we see edicts issued in multiple languages and once the Vikings come in and the Normans take over all bets are off as English changes drastically. The language of the Angles, Saxons, and Jutes although handy ultimately only served as a transition as even Old English ultimately is going to undergo dramatic changes in the next 600 years from the invasion. This ultimately makes it hard to distinguish between the various additions and when the particular language of the Britons was ultimately abandoned in England. And in places it didn't! If you look at the area of Cornwall Cornish survives to this day! This is ultimately the descendant of one of the Brittonic languages of the tribes!

Moreover in Britain the linguistic issues have almost always led to the conqueror's language gaining dominance. Look at the Norman conquest and the addition of French to English. Even in Scotland Gaelic uprooted Pictish and then the Vikings added in with Norse. It led to the Isles of Orkney speaking Norn a Norse hybrid well into the 1700s!