The first Saxon settlers were retirees from the Roman Legions, which often had Saxon contingents. Saxon retirees stayed in Britain when the Legions pulled out. An English King was so impressed by the Saxons’ fighting skills he hired them to help ward off Pictish invaders. Those dastardly Scots were evidently a greater threat to the English.
The Saxons decided England was not a bad place to stay and invited their friends to come over, just like one of those mad Melbourne parties you read about where gate crashers terrify everyone and the police are outnumbered. The next day everyone wakes up with a headache and promises never to drink again… or mess around with Saxons armed to the teeth.
However recent DNA testing results are showing that maybe only 6 - 8% of the English population can trace their ancestry to those evil Saxons, so perhaps the English are not Anglo-Saxon after all?