Anglo-Saxon monk Bede lived from (673-735) and is hailed as the first English historian. He's our only contemporary source of the time, and wrote just before the beginning of the Viking age. He wrote fairly extensively on Anglo-Saxon origins in his 8th century works The Ecclesiastical History of the English People. Bede states:

Those who came over were of the three most powerful nations of Germany ­ Saxons, Angles, and Jutes. From the Jutes are descended the people of Kent, and of the Isle of Wight, and those also in the province of the West Saxons who are to this day called Jutes, seated opposite to the Isle of Wight. From the Saxons, that is, the country which is now called Old Saxony, came the East Saxons, the South Saxons, and the West Saxons. From the Angles, that is, the country which is called Anglia, and which is said, from that time, to remain desert to this day, between the provinces of the Jutes and the Saxons, are descended the East Angles, the Midland Angles, Mercians, all the race of the Northumbrians, that is, of those nations that dwell on the north side of the river Humber, and the other nations of the English.

Bede is writing a good couple hundred years after the event, and draws much from a Sermon written by a Briton called Gildas. Both Gildas and Bede paint a picture of violent invasion and enslavement, though archaeological evidence overwhelmingly points to a more gradual and peaceful settlement.

In the 9th century, the Anglo-Saxon Chronicle is created, which tells again the story of the Anglo-Saxon settlement of Britain, and contains text from Bede's Ecclesiastical History.

From these sources, we can be certain that at least the learned elites and monks knew their origins. However, as with most Medieval sources, it would be impossible to tell what the average Saxon peasant knew about their history. The vast majority of peasants were almost certainly illiterate and only monks and some of the elite would have been literate.

Sources:

St. Bede, Jane, L.C. (trans.) (1901), The Ecclesiastical History of the English Nation, New York, Cosimo.

Miles, D. (2006), The Tribes of Britain, Revised Edition edn, Phoenix, London p.164.

Härke, H. (2011), Anglo-Saxon Immigration and Ethnogenesis, Medieval Archaeology, vol 55, p. 16-17.

Follow up, did the Vikings realize the connection either?

As someone asked but then deleted their comment, to clear up any confusion:

Angles came from Schleswig, Saxons also came from Schleswig but from the Baltic Coast. The other people who invaded the isles during this period were the Jutes, but their original location is unknown exactly but theorized to be Jutland.

As you can see here Schleswig is the center of Jutland which is Denmark's main peninsula.

Viking raids came from modern-day Denmark/Norway/Sweden, which Schweslig is right in the middle of. You can see a map of 8th century settlements before they began to expand here

Edit: Some have noted this is a more simplistic view of what actually happened. Here's a good source: https://www.reddit.com/r/AskHistorians/comments/3qyzra/how_significant_was_the_celtic_population_of/

To be fair, wouldn't it be more correct to say "the place where SOME of their ancestors migrated from"? There were Frisians and Saxons after all.

Was there any reason why the invaders left their homeland?