My theory is that the Anglo-Saxon speakers literally displaced the Celtic speakers; they were either killed or ran off to places like Wales and Scotland. This is sheer conjecture on my part, though. I don't know enough about that part of history to be sure.
This was the standard theory until recently but apparently new studies have shown that there was much more continuity in the population than was previously thought. The book I was reading about this is at home and I'm at work so I'll get more info tonight.
Most of the acknowledged
Celtic-derived words in English are placenames and words for topographical features. Avon, tor, lough, and so on. Again, my list is at home.
And where do the Celtic languages come from? I assume that there was a heavy Viking influence, but I don't even know that for sure.
In the beginning (well not quite but close enough) there was Indo-European, probably spoken in what is now the Ukraine or Southern Russia. With migrations, conquests and so on Indo-European broke up to form other languages, just as the Romance languages developed out of Latin. One branch became Sanskrit and the languages of Northern India. Another branch became Iranian. Another branch became Greek. Another branch became the Italic languages, including Latin. Another branch became the Germanic languages (including eventually Old English, Old High and Low German, Old Norse, Gothic, and so on), and another branch became the Celtic languages.
The earliest Celts we know of lived on the Danube in the 5th century BC but spread out into most of what is now Western Europe. The Romans knew them as Galli (Gauls). Italy north of the Po was called Cisalpine Gaul (Gaul this-side-of-the-Alps). In the third century BC Gauls were used as mercenary troops by some of the Successor Kings who fought over Alexander the Great's empire. Some of these troops settled in parts of modern Turkey and were known as Galatians (hence Paul's letter to the Galatians). There seem to have been two main migrations of Celtic speakers into Britain. The first wave (4th century BC?) reached Scotland, the Isle of Man, and Ireland. Gaelic, Erse, and Manx developed from the language of this wave. A second wave settled in Southern England and Wales. Their language later developed into Welsh, Cornish, and Breton. The Romans conquered the Gauls. As the Germanic tribes took over the Western Roman empire in the 5th century, they took over the Gauls as well. In continental Europe the Celtic languages died out except in what is now Brittany, where the Celtic language was kept going by refugees from Britain (the similarity of name is no coincidence). As I said above, there seems to have been some continuity of population, but the Celtic languages have only really kept going in the far west.
The Vikings spoke Scandinavian languages, which are part of the Germanic branch of Indo-European. Starting from the 8th century AD, Vikings from Denmark, Norway and Sweden moved out raiding and trading everywhere their boats could carry them. One lot, the Rus, went down the rivers from the Baltic to found Russia. Another lot came to N. France to found Normandy (North Man dy). Many Vikings settled in Eastern England (the Danelaw) and this had a huge impact on the English language. There is one theory that says that this is when and why Old English started losing its inflections. The stems were similar enough in Old English and the Viking languages, but the grammatical subtleties of inflections were too different, and so the two groups had to find other ways of making themselves understood.
Then came 1066. The Normans (Vikings) who had settled in N. France had taken to using the local French dialect rather than keeping their own language. So that was what they brought over to England with them, resulting in the Germanic-Romance hybrid we know and love today.