Like most topics in Late Antiquity, the "Adventus Saxonum" is quite complex.
Well, first, let's clarify some termonology: Saxon was the generic Roman name for many different groups raiding the coasts of France and Britain from the 3rd-5th centuries AD. Culturally and genetically, recent research as shown the majority of the 3rd-5th century "Saxons" were actually Frisii, or Frisians. The Saxons themselves really ramp up the raiding after the collapse of the West, but in terms of the terminology used by the sources, it really refers to multiple peoples during the 5th-7th centuries who were part of at least one, probably more, tribal confederations.
As for their choice to attack Britain, the Saxon invasion of Britain is still open to much debate. Sources like Gildas' "De Excidio Conquestu Britanniae," Bede and his "Historia Ecclesiasticae Gentis Anglorum", and the "Historia Brittonum" claim that the Saxons were invited as Foederati - or regiments of soldiers serving the Romans. The early Saxons were given settlements in Britain and then landed in exchange for their military service. Regiments of Saxon Foederati do pop up in the sources - Frisians had been serving on Hadrian's wall since the 1st Century AD, and some served Aetius at the Catalaunian Fields in 451, and Saxons and Franks are believed to have comprised significant portions of Riothamus' army in 469-470. Archaeolgically coin hoards of Valentinian III extend as far north as Oland, showing Aetius was recruiting significant numbers of Germanics from the North Sea and Baltic region, probably as a result of Hunnic treaty conditions. However, whether the "revolting foederati" hypothesis is true isn't certain since other sources paint a different picture.
Contrary to popular belief, Roman Britain was not "abandoned" in 410 AD. That's based on the "Rescript of Honorius" which is believed to be addressed to the city of Bruttium in South Italy, at the time under threat from Alaric/Athaulf and the Visigoths. Archaeologically, Much of Britain remained under Roman control until the 430's, and areas of South Britain are archaeologically consistent with the Roman continent until about 450.
The first account of Saxons in Britain comes from Constantius of Lyon's "Vita Sancti Germani" where it is recorded he was sent on a mission to stop the spread of Pelagianism in the Island, and used his military experience to lead some British forces to victory when an army of Scottii, Picti, and Saxones was threatening the region of Wales. Whether or not this is real, a later addition, or simply attributing it to divine providence is unlikely. The Gallic Chronicle of 452 is next stating that the region had been "reduced to Saxon rule" in 441, and then of course there is the famous "Groans of the Britons" to Aetius dated to 446 in Gildas. As a whole this is consistent with the archaeological evidence where noticeably Saxon artefacts (e.g. Fixed Plate Dragon Buckles) appear around the mid 5th century. This is also around the time that the warlords Vortimer and Vortigern are alleged to have existed, although the history of the figures surrounding King Arthur is a whole other debate.
From what we know of other barbarian groups, since the Saxon invasion like much of the 5th century is poorly attested, it seems the invasion and the federate hypothesis are both valid. These peoples did not constitute a single political grouping, but were more like complex confederations. Saxons would have been serving the Romans at the same time as they were serving themselves, and federate groups often went rogue (e.g. some of Aetius' Huns did in South Gaul and had to be put down by Avitus). Their invitation and service may have been on the part of the Roman government but also may have been on the part of local aristocrats. Bryan Ward Perkins compares the collapse of Britain to that of Noricum from 454-476, where the administration simply didn't end, it slowly broke down into effective city-states and barbarian groups slowly moved in, and Roman soldiers stayed at their stations until they realized the pay had stopped.
Now, as for why the Frisians, Saxons, etc. chose Britain? They had been raiding there for some time and there were other Barbarian confederations (The Franks) in the way of attacking Gaul, and the Hun empire was also to the South. But, however, it is known there were Saxon and Frisian settlements in other areas - e.g. Flanders, and around Bourges. These peoples picked targets of opportunity at times of Roman weakness. But to invade and conquer Roman Gaul would have meant conquering the Franks on the upper Rhine, who were organized and also in Roman service at the time (usually). Britain had material wealth and resources - and the Romans didn't have the resources to commit to its defense other than what was already there (roughly about 20,000 Limitanei spread across Hadrian's wall and down the shores). It was accessible by boat and these peoples were known to be seaborne merchants and raiders (it is often forgotten not all exchanges were hostile), and being recruited by the local aristocracy was a lucrative means to acquire wealth. Permanent settlement was another matter as military service would not have included women and children on these expeditions (usually, there's some evidence women were present in the first wave of viking expeditions for permanent settlements, for comparison). If they were given land grants, Women and Children may have come across later, and the warriors would probably have kept locals for wives as well (I'm not gonna get into the genetic evidence for the invasions though, that's something I don't know enough about to elaborate on). However, it seems that by the mid-5th century there were permanent settlements of these peoples in Britain. The number of immigrants is up for debate (again I'm not gonna get into Genetics here), but archaeological evidence suggests it may have been as few as 20,000. (The number is still debated). This settlement was not concentrated. The first princely graves don't appear until 600 AD, so their ability to excise political power was limited before then. However, it was extensive enough to promote Saxon acculturation over Romano-British acculturation in the settlement regions, despite the fact that the majority of the population was not Saxon. There was a presence of Saxon aristocrats (probably leaders of Manus or warbands) and they were able to excise a controlled authority over their followers and over the regions they controlled. The presence of large kingdoms would not come until as I said, much later in the late 500's and early 600's. As I already said, it wasn't one single invasion but several immigrations and intrusions over the course of 2-3 centuries, which resulted in a complex political evolution and ethnogenesis.
Sources: Bohme, "Das Ende der Romerherrschaft in Britannien und die angelsachsische Besiedlung Englands im 5. Jahrhundert" 542-548 (check out these pages about Roman and Saxon 5th century archaeological sites: http://www.fectio.org.uk/articles/hwb/hwbindexfigures.htm)
Guy Halsall, "Barbarian Migrations and the Roman West" (For a complex overview of the Migrations as a whole)
Bryan Ward Perkins, "The Fall of Rome and the End of Civilization" (for comparison between the collapse of Roman Noricum and Roman Britain, although I should note a lot of authors disagree with some of his work in here).
Marcin Buczek, "Frisians in Roman Britain in light of the Available Epigraphic Sources."
Roymans, "Gold, Germanic Foederati, and the End of Imperial Power in the Late Roman North," in Social dynamics in the Northwest Frontiers of the Late Roman empire. Beyond decline or transformation, 60-65.
Gerrets, "The Anglo-Frisian Relationship seen from an Archaeological Point of View," in Freisishe Studien II (1995), 119-128
Garrett, "Non-Legionary troops in Roman Britan: Part One, the Units," in Britannia 25 (1994): 60.
The Anglo-Saxon Cemetary at Spong Hill (For estimate of settlers, free here: http://eaareports.org.uk/publication/report69/)
Dark, "Civitas to Kingdom: British Political Continuity," 300-80 and also Hines "Philology, Archaeology, and the Adventus Saxonum vel Anglorum" 17-36. (On the sporadic initial adventus Saxonum and archaeological evidence during the Mid 5th century)