Good question. I should probably make a preliminary note that the sources for this period and the history of the Saxon people in "Germania" is somewhat scant. You have the writings of Tacitus and Pliny covering the 1st and 2nd centuries (with their usual Roman-style caveats) and then a patchwork of information leading up to Saxon migration in the 5th century. After that, you are relying on Saxon, or even Byzantine (Procopius), sources to paint a picture of post-Roman England and the migrations.

It will probably come of no surprise that Saxons were used as foederati by the late Roman Armies and it is likely some were already in Roman Britain prior to 410. They had an awareness of the land, its people, and culture which was brought back to their holdings in northern Germany and beyond. To use the modern term, this was an inviting "pull factor" for migration.

Secondly, the climate provides a sizeable "push factor" for the Angles, Saxons, and Jutes. Occupying much of the modern German state of Holstein, the "Cimbrian Peninsula" of modern Denmark, and down towards Holland, they made their home around the narrow seas and deep inlets of this section of Northern Europe. Yet with a warmer climate than normal and the ice caps melting, these were lands prone to flooding. A terrible prospect for a society quite grounded in agriculture. This rise in sea levels is likely to have provided more navigable routes around the east coast of England which were perfect for Saxon boats to penetrate the land from The Wash in East Anglia and down its many tributaries.

Finally, it is important to analyse the geopolitical ecosystem around the Saxon realms. If the Saxons were to migrate outwards, it was Gaul, not England, that would naturally have been in their sights. Gaulish heartland was a richer and nearer province and far away from the Litus Saxonicum per Britanniam (a network of forts and defences on the South Coast of England and the periphery of Northern Gaul built by the Romans which still posed a threat to any invading force). Even decades after the Roman abandonment of England, the Saxons were still trying to gain a foothold in Gaul and did manage to capture Angers in 463. However, the Franks were slowing flexing their muscle to chase out the Saxons of Angers and eventually the last remnants of Roman Gaul ("The Kingdom Soissons") in 486 making them true masters of the region. One historian makes a particularly agreeable point that with Saxon confidence shattered by the Franks, an "easier more familiar" target was preferred.

With few other areas to turn to for expansion, the island to the West proved a worthy choice free from climate change and a relatively familiar land for some which were easily accessible to these maritime people. From this point, it is worthwhile looking into how Saxon dominance of the land was established and how the natives reacted to Saxon culture and the elites and the effects on both sides after the Battle of Mount Badon circa 500AD.

Sources:

F Stenton, Anglo-Saxon England (Oxford 1971)

P Salway, Roman Britain (Oxford 1981)

H Lamb, Climate, History, and The Modern World (New York 1997)

Note: Instead of saying "Saxons, Angles and Jutes etc" every time I have just used "Saxons" for ease.