We often see Britain as a separate entity, standing apart from Europe, with the Channel forming a natural barrier. However in Roman times that was not the case - of course, the Channel was there, but it was not the barrier it is today. Back in the day the sea was the main way different parts of Europe were connected. Communication and trade over land was difficult - certainly before the Romans built their roads. But even with Roman roads, the sea still facilitated easier and quicker transport of goods and people.
Since the Iron Age there has been regular contact and trade between Britain and mainland Europe. We know that with the Roman expansion in southern Gaul trade between (southern) Britain and Roman territories became regular, though it likely didn't amount to a large volume (part of the trade had to take place over land to western/northern France). Ores and slaves from Britain were traded for Roman wine.
It is in this context that the first Roman invasions of the Britain took place. As he described in his "Commentarii de Bello Gallico", Julius Caesar had noted that Gallic tribes in what is now northern France had many ships and that they regularly sailed to Britain. Moreover, Caesar knew that the Gauls he had been fighting had received reinforcements from Britain (considering the number of Gallic coins from this period found in southern Britain, they were likely mercenaries). Caesar used the latter as the reason the justify his invasions, but no doubt that gaining control of the source of valuable trade goods played a part in his decision making.
Caesar's invasions had no lasting success, but they did have a lasting influence. Some tribes in Britain developed strong ties with the Romans - helped by the fact that Roman territory was now just across the Channel. Embassies were sent to the new emperor and there's evidence of quite significant numbers of Roman merchants living (!) in Britain during this time.
In 43 AD the Roman emperor launched the successful invasion of Britain. His exact motivation is unknown but the general consensus is that he needed a war - and a victory - to establish his legitimacy as new emperor. Roman society put a lot of stock in military prestige. To be sure Britain will have looked as the most tempting target, considering the major setbacks Rome had suffered trying to expand across the Rhine. The invasion was made easier by the fact that his predecessor Caligula had actually made preparations for an invasion. Claudius personally led the invasion and did manage to become a pretty strong emperor as a result.
A couple of books covering this and a lot more are "Roman Britain" by T.W Potter and Catherine Johns and "The Romanization of Britain" by Martin Millett.