Good question, and one for which there is no general consensus - it's actually a hotly debated topic. Broadly, there are two opposing viewpoints. One position is that there was a mass folk migration of Anglo-Saxons by which most Britons were displaced or enslaved. The other is that a much smaller number, a warrior elite, invaded and established themselves as rulers, similar to the Norman Conquest, except that the native Britons adopted the incoming culture (including the language) and gradually "became" Anglo-Saxons. Areas which weren't settled by the Anglo-Saxons (such as modern Wales) retained their British identity. Most of the evidence from both sides of the argument comes from non-literary sources, as the Anglo-Saxons weren't literate until after their conversion to Christianity which began around 600 AD, so all the literary sources which deal with the migration period are far too late to be reliable with regard to the fifth century.

Here is a recent study on the genetics of various human populations: http://www.sciencemag.org/content/343/6172/747

This data suggests that there is no strong genetic evidence of admixture in the UK population.

Thus, the number of Angles/Saxons/Jutes who migrated to Britain was not significant enough for them to leave a strong genetic heritage in the modern British population, i.e. either few of them arrived, or their genes were selected against (the former is more likely).

Edit: duplicated words.

More edits: Play with the data yourself, here: http://www.admixturemap.paintmychromosomes.com/

Non-sciency write-up:

http://popular-archaeology.com/issue/12012013/article/scientists-create-genetic-map-of-history