The Angles, Saxons and Jutes arrived in southern Britain in the 6th and 7th centuries. It's a matter of historical contraversy how (cultural assimilation or invasion?), but essentially they somehow ended up in charge, with the native Britons underneath them. While they probably weren't a big part of the population, their cultural dominance essentially turned the Britons in that area into Anglo-Saxons. The area they controlled became known as "Angle-land" or later, England. It comprised approximately what we now call England. http://orig03.deviantart.net/92a0/f/2014/313/d/4/anglo_saxon_migration_by_arminius1871-d85tqiw.png

The language of the Anglo-Saxons was what we call Old English. It was later influenced by the Viking conquest of northern England, though not massively so.

Now, in 1066 William the Conquerer became King of England. He and his men spoke French (or at least, various northern dialects of French) and so for about 300 years, the language used by the King and in courts was actually French, not English (records of course, were written in Latin, like in most of Western Europe). However, in the 14th century, the status of French declined, slowly being replaced by English. In 1390, Henry IV became the first King since before William's invasion to take his oath of Kingship in English. French continued to decline, being replaced by English. Though a few French phrases do remain in government usage (for example, British passports have an Old French phrase on the front, which is the motto of the Queen). By this time, England had essentially conquered Wales, and when Henry VIII was King, he unified the legal systems, and thus Wales had the same official language (of course, nowadays Welsh official documents and signs are all in both languages, as both are official languages).

I'm afraid someone else will have to tell you the history of Scots.