I don't think anyone claims there was no migration, just that there appears to be no definite identifiable point at which a mass migration or large scale invasion happened. Michael Pye's The Edge of the World is an interesting read relating to this topic. "Frisian" trade networks spanning the North Sea coasts may be a big part of the (hypothetical) answer. There are lots of references to the Frisian sea, Frisian trading posts on all North Sea coasts, and Procopius (from a vantage point in Byzantium) writes that Brittia is inhabited by Angles, Frisians, and Brits.

"Frisian" as used by Latin writers in the Dark Ages should be understood as a purely geographic label: the oldest written history of the Counts of Holland mentions that the Low Countries are inhabited by Saxons, but are traditionally called Frisians by the Romans and Franks (after a tribe that used to live there in Caesar's time that gave the area its Latin name). So for practical purposes Frisians = Saxons. One is an endonym, and the other an exonym for ethnically the same people in a specific area.

Because transport over land was much slower, harder, and more dangerous than transport over sea, this trade network would have had a major impact on linguistic transmission (creating perhaps a sort of creole Lingua Franca of the markets on the coast), and migration from coast to coast was simply a matter of individuals, families, small bands of adventurers, or small villages booking passage over the course of centuries. If migration happened this way, the migrants would easily all pick up the same language (closely related to their own). And diplomats traveling between the kingdoms in England would have depended on that same trade network for their travels. Travel itineraries of missionaries for instance do suggest that hopping from port to port on the North Sea was greatly preferred over inland travels.

Kingdoms that formed may have picked this trade language as their official language merely as a matter of convenience. It was the language of wealth and power, and of interaction with the other kingdoms and foreign bands of cheap mercenaries. Like the US picked English, over for instance Dutch, French, Spanish, or Navajo, etc. Including some kingdoms that were, by historical accident, dominated by clans of Angles.

This account leaves open the question of what happened with the trade network if it was already so well-developed. How did English become isolated from the mainland Germanic languages? Another hypothetical: The Franks caused it to collapse when they conquered the Frisian and adjacent Saxon kingdoms on the mainland, temporarily isolating Britannia and the Scandinavian coasts from access to the trade network, and making it give way to an era of North Sea raiding (the "Viking" era) that pushed people away from the coasts and increasingly turned inland travel into the preferred method, and reduced shipping to short distance crossings of the English Channel.