This is indeed a very striking fact of the Anglo-Saxon conquests. The linguistic evidence is particularly stunning; at present, we can only really find a handful of words from a British substrate in modern English, which is low even for a substrate language. Even more impressively, we have a model for what is on the surface a contemporary similar relationship—the Frankish conquest of Gaul—where the invaders were successfully assimilated, with a thin layer of superstrate political vocabulary grafted onto the adopted language.
The reason for the radical difference between the British and French case can most likely be found through evaluating why people adopt other language. Fundamentally, it's an issue of prestige: if an invader conquers an advanced civilisation with sophisticated structures of government and a language of administration to accompany it, they are much more likely to assimilate into the superior culture than to reject it and supplant it with their own. Conversely, a subject people is more likely to adopt the language, culture and values of their overlords if they provide a coherent and socially superior model to their own.
So what did Britain look like when the Anglo-Saxons arrived? Practically speaking, it was not in good shape. The Romano-British as a coherent culture and society had rapidly eroded by the turn of the 5th century, with the occupation of sites having collapsed, and activity that re-used Roman sites appearing not to be a fundamental continuation of the Romano-British tradition. In France, on the other hand, there was significant, though uneven, continuity of government, culture and political classes.
As Britannia had been thoroughly deromanised, then, Anglo-Saxons simply did not feel the need to assimilate; the Franks, however, had significant pressure to do so. Thus the Germanic languages were more appealing to the British natives, whilst Latin/Gallo-Romance was more appealing to Franks.
Sources:
B. Ward-Perkins, "Why Did the Anglo-Saxons Not Become More British?", The English Historical Review, Vol. 115, No. 462 (Jun., 2000), pp. 513-533
N. Faulkner & R. Reece, "The Debate About the End: A Review of Evidence and Methods", The Archaeological Journal, Vol. 159 (2002), pp. 59-76