The Romano-British inhabitants of sub-Roman Britain inherited Roman laws, organizational structures, and military tactics, yet they were unsuccessful in repelling the invasions of the Anglo-Saxons in the fifth and sixth centuries AD. And unlike the rest of the Roman provinces overrun by Germanic tribes, the Britons had a natural barrier in the form of the English Channel. Besides the (albeit of dubious historicity) Battle of Badon Hill where King Arthur stemmed the tide of invasion, the period is characterized as a constant press of Germanic invaders forcing the Britons out of the prime English land. What made the Anglo-Saxons so successful? Is there evidence of a disparity in numbers, tactics, weapons, or some other factor?
It's not certain that they conquered anything. Exactly how what we now call Anglo-Saxon culture came to dominate England is a matter of considerable ongoing debate and study. Basically, the evidence we have is a mixed bag. There's basically three broad schools of thought:
Invasion and takeover. The traditional take, found in written sources like Bede's Ecclesiastical History of the Britons and Gildas' De Excidio et Conquestu Britanniae. Germaic mercenaries were invited into Britian to help the Romano-British fight off Pictish and Celtic raiders, decided to take the land for themselves, invited their tribesmen to join them, and ended up driving the Romano-British westward and northward into what is now Wales and Yorkshire.
"Small bands" theory. Small bands of warrior elites did enter into Britain and oust some existing leaders. Most of the rest of the population, however, stayed right where they were, and simply adopted Germanic cultural practices, including language, to get on the good side of their new masters. Over a couple of centuries, these gradually became dominent. Similar to the Norman invasion, or the early history of Russia. Some recent genetic studies of the British population tend to support this theory.
Gradual acculturation/integration which simple accelerated after the end of the Romano-British period. In this view, there have been strong trade ties and evidence of population exchange between Southern Brittain and parts of the continent since the late Iron Age, and a form of proto-English may already have been in place at the time of the Roman invasion. With the political collapse of the Western Roman Empire, these ties deepened, and the indigenous, Germanic-descended culture which is know to us as Anglo Saxon gradually took power over the course of a few centuries. Some genetic and archelogical evidence for this.
[1]Russell, Charlotte. The Anglo-Saxon Influence on Romano-Britain: Research past and present. [2]Härke, Heinrich. Anglo-Saxon Immigration and Ethnogenesis. Medieval Archeology, Vol. 55, 2011.