The great conspiracy of 367 ad
The Saxon migration to Britain appears to not be something that happened overnight and instead appears to have happened in stages.
Perhaps the first trace of Saxon raiders showing an interest in Britain can be traced back to the great conspiracy of 367 ad, as documented in "The Roman History of Ammianus Marcellinus".
The Roman History of Ammianus Marcellinus, 8-5
5 It will, however, be in place to say, that at that time the Picts, divided into two tribes, called Dicalydones88 and Verturiones, as well as the Attacotti, a warlike race of men, and the Scots, were ranging widely and causing great devastation; while the Gallic regions,89 wherever anyone could break in by land or by sea, were harassed by the Franks and their neighbours, the Saxons, with cruel robbery, fire, and the murder of all who were taken prisoners.
There was another reported wave of attacks in 409-410 AD, according to many Wikipedia sources, when Constantine III was hugely criticized for leaving Britain defenceless during his march to Italy to remove Honorius from power in Rome, and apparently Germanic Saxon tribes took advantage of this and began raiding and pillaging Britain.
In 409 Constantine's control of his empire fell apart. Part of his military forces were in Hispania, making them unavailable for action in Gaul, and some of those in Gaul were swayed against him by loyalist Roman generals. The Germans living west of the Rhine River rose against him, perhaps encouraged by Roman loyalists, and those living east of the river crossed into Gaul. Britain, now without any troops for protection and having suffered particularly severe Saxon raids in 408 and 409, viewed the situation in Gaul with renewed alarm. Perhaps feeling they had no hope of relief under Constantine, both the Romano-Britons and some of the Gauls expelled Constantine's magistrates in 409 or 410. The Byzantine historian Zosimus (fl. 490s – 510s) directly blamed Constantine for the expulsion, saying that he had allowed the Saxons to raid, and that the Britons and Gauls were reduced to such straits that they revolted from the Roman Empire, 'rejected Roman law, reverted to their native customs, and armed themselves to ensure their own safety'.
Matters were made worse for Britons when in 410 AD Honorius decided to cut his losses with Britain and refused to send foreign aid, according to Wikipedia.
Stilicho and the defense of Italy, Wikipedia
By 410, Britain was effectively told to look after its own affairs and expect no aid from Rome.
By 441 AD the Saxon migration to Britain must have been well under way because a greatly disputed quote attributes the British provinces to being subject to Saxon rule as early as 441, as documented in the book Celtic culture
Groans of Britons
According to Wikipedia, it is written by both Gildas and Bede that yet more Saxons arrived on British shores during the Groans of Britons when the Brits made a final plea to Rome for help which was denied, and this led the Brits to invite Anglo-Saxon mercenaries in to Britain to help deal with Scots and Picts that were raiding the Southern lands.
The message is recorded by Gildas in his De Excidio et Conquestu Britanniae, written in the second quarter of the sixth century and much later repeated by Bede. According to these sources, it was a last-ditch plea to "Agitius" for assistance. Agitius is generally identified as Aetius, magister militum of the Western Roman Empire who spent most of the 440s fighting insurgents in Gaul and Hispania. The Roman Britons had been beset by raids by the Picts and Scots from northern Britain, who were able to pillage far to the south after the Roman armies had withdrawn from the island in 407.
Why did so many Saxons emigrate to England?
According to the World History Encyclopedia the Franks already had a strong presence on European mainland making expansion difficult for the Saxons. Add this to the fact that Britain was left defenceless by Constantine III, and the Saxons were presented with the perfect opportunity to instead expand in to Britain.
Saxons, History, World history encyclopedia
South of the territory where the Saxons lived on the continent were the Franks, a strong Germanic confederacy that had a solid presence occupying a territory between the Saxons and the Roman frontier. For this reason, expanding southwards was a problematic option for the Saxons, and a sea expansion was a more suitable alternative.