Roman Britain certainly wasn't isolated from the rest of Europe, keep in mind that in times before the automobile and planes, boats and travelling on foot were the norm. A boat ride wouldn't have taken nearly as long or as tiresome a trip as walking somewhere.

There was constant trade between the British isles and Europe long before the Romans ever arrived, actually after Caesar had conquered Gaul the British were trading with the Romans.

However to get back to your question, the Anglo-Saxon invasion of Britain is a bit more complicated than a bunch of foreign barbarians landing on the British shore and taking over.

The Saxon Shore was a defensive line of Roman forts that spanned both sides of the English channel to protect from "barbarian" raids. These raids came mostly from the Saxons which is why the name Saxon Shore stuck, but they weren't alone in their raids. These defenses were erected sometime during the late 3rd century, and did a fairly good job of defending the Romano-Briton and Romano-Gallic civilians on both sides of the channel. However during the 4th century crisis that the Roman Empire faced, and under the rule of Magnus Maximus the Roman legions were withdrawn from Britain around 383 to help defend the mainland borders from Migrating tribes.

It was during this time when the Roman Empire withdrew from Britain that the Romano-Britons were left to their own defenses. Strangely the Britons looked to the Germans who were raiding their shorelines to help defend against other raiders such as the Picts, Scoti and other Saxons by promising them gold or land as payment for their services as Foederati.

While the details on what exactly caused issues to arise are a bit fuzzy the relations between the Romano-Britons and the Anglo-Saxon Foederati soured quickly. Maybe it was over land and the Foederati wanted more for their crops and families, or maybe the Britons didn't have enough gold or food to pay the Foederati. Either way things got bad and an Anglo-Saxon revolt broke out.

In the aftermath, or maybe during more Angles, Jutes and Saxons came across the North Sea and English channel in three or five small invasion fleets. Whether they were called by the Anglo-Saxons already on the British isles or if they just came of their own volition we're not entirely sure and it's still left to debate by historians.

The Angles, Saxons Jutes and Frisians had a complicated relationship with the Romano-Britons. From raiders to defenders, and even subjects to them, then finally invaders. It's certainly a hard relationship to make sense of when we have so little information left from around this period.

References:

Bede. Ecclesiastical History of the English People.

Ward-Perkins. The fall of Rome: and the end of civilisation Particularly

Myers. The English Settlements, Chapter 5: Saxons, Angles and Jutes on the Saxon Shore

Hopefully this is a good answer to your question.