I find the "why didn't the Senate stop him" part of your question very interesting. The answer is that they weren't allowed to. The Senate had given Caesar a mandate to govern Gaul as proconsul for a certain period. That mandate could not be revoked except under very extreme circumstances. The only choice the Senate had regarding Caesar was whether or not to renew his mandate once the set period was nearing its end. Their refusal to renew Caesar's mandate sparked the civil wars which ended the Roman Republic.

Imperium

Let's go into Caesar's mandate a bit more. As proconsul, Caesar held imperium. Back when Rome was a small kingdom, imperium was the the authority of the King to act in the public interest. As the authority of the King, imperium was indivisible. One person had the power to do whatever was necessary, with only minimal safeguards against abuse of power.

When the Romans deposed the king and Rome became a republic, they devised a system which prevented too much power from concentrating in any one person. In this early system, every year the Senate would elect two consuls. Each consul would have imperium and each would command half the army. Each consul had the power to veto decisions made by the other consul. After a year the consuls would step down and two new consuls would be elected. With the power spread between two persons and with no one holding imperium for more than a year, it would be harder for one person to have too much power.

There were some additional safeguards added over time. People's tribunes had the authority to veto the decision of any magistrate. Furthermore, in times of crisis the Senate could appoint a dictator who would singly receive imperium for six months, replacing the two consuls. This happened for instance when Hannibal had defeated a Roman army at Lake Trasimene, killing the consul who led that army. The Senate appointed Quintus Fabius as dictator. After raising a new army Fabius joined the remaining consul Servilius, relieved him of command and merged the two armies to defend Rome against Hannibal. A dictator was only appointed when Rome itself was under threat though, so it wouldn't be considered as a measure to stop Caesar from invading Britain.

The extension of imperium

In the fourth century B.C. a new role was added to the political system. In order to reduce the work load of the consuls, the Senate started appointing praetors. A praetor would also have imperium, but of a lesser degree than the imperium of consuls. Initially every year one praetor was appointed, his primary role being the administration of justice. In the third century this was increased to two praetors, one being responsible for Roman citizens (the Praetor Urbanus) and the other for non-citizens (the Praetor Inter Peregrinos). With the institution of the praetors, consuls would only have to concern themselves with justice in extraordinary cases.

As the Romans increased the territory they controlled, more magistrates were needed. Halfway through the third century two more yearly praetors were added, one to govern Sicily and one to govern Sardinia. From 197 (B.C.) onwards, Spain was divided into two provinces governed by praetors. Another measure the Senate took from the third century onwards, was to allow consuls to continue to hold imperium for another year. The Senate would still elect two new consuls, but one or both of the previous consuls would be allowed to operate as proconsul ("acting in place of a consul"). This meant that he could continue whatever he was doing (usually fighting a war) while the new consuls would receive other tasks. Such an extension might happen several years in a row. Later on the same would apply to praetors, allowing some to stay on as propraetor.

By the first century B.C., the Senate would under normal circumstances appoint two consuls (with full imperium) and six praetors (with lesser imperium) every year. Additionally, they could extend the imperium of sitting consuls and proconsuls for a year, as well as the lesser imperium of praetors and propraetors. Though initially imperium was not tied to a geographical area, over time it did evolve to a point where someone was understood to have imperium not only for a limited time, but also over a limited area. This is the basis for the provinces governed by proconsuls and (pro-)praetors.

Back to Caesar

Thanks to political maneuvering, Caesar had managed to get his proconsulship in an unprecendented five year chunk (instead of the usual yearly extensions). As proconsul, Caesar governed Gallia Cisalpina ("Gaul this side of the Alps"), which was the part of modern-day northern Italy between the Rubicon and the Alps, and the Gallia Transalpina ("Gaul the other side of the Alps"), which was modern-day southern France from the Mediterranean to whereever Roman rule ended - at least when Caesar was assigned these provinces. Through relentless warfare, Caesar extended Roman rule farther and farther north. Since there was no well-defined northern edge to his geographical area of responsibility (as there was with the Rubicon in the south), there was nothing stopping him from going on and on. The only time when the Senate could have stopped him, if they wanted to, was after the first five years were up. Caesar arranged for his proconsulship to be extended for another five years though, and launched two invasions of Britain. When Caesar learned the Senate would refuse another extension of his proconsulship, he marched his army to Rome and started a civil war.

Summary

The Senate had granted Caesar imperium as proconsul. This imperium meant that Caesar had free reign to do as he wanted, with very few limits. The way the Roman political system had developped, power was mostly limited by the yearly elections/renewals, but Caesar had worked around that by getting his appointment five years at a time.

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Sources (the couple most interesting)

• T. Corey Brennan, Power and Process under the Republican "Constitution", in: Harriet I. Flower, The Cambridge Companion to the Roman Republic (New York, 2004) p. 31-65

• A. Lintott, The Constitution of the Roman Republic (Oxford 1999)

(edit: annoying spelling mistakes)

I'd like to add on to /u/edXcitizen87539319's perfectly correct response by saying that the Senate was also nothing more than an advisory body. The only thing the Senate actually did was appoint magistrates by overseeing their elections. The Senate did not have any official power--that was handled entirely by the magistrates. Now, granted, the Senate could always issue their consulta, most famously their senatus ultimum consultum (the name is a modern coinage). And several times they did this during Caesar's lifetime. But even the ultimum consultum had no actual legal force behind it. No one was actually required to follow any of the consulta, you just did so anyway because your career would be in the toilet if you didn't and every person in Rome would despise you for ignoring the voice of the majority.

The other thing is that the degree of success of Caesar's British campaigns are debatable. His first campaign was in no way an invasion. He sent over a paltry number of troops to an island that nobody had ever been to. His first campaign was very much a scouting expedition. Now, of course, every time Caesar goofed up he was apt to call it a scouting expedition (when the Germans gave him trouble--even though he technically won--across the Rhine, for example), but inthe case of his first British campaign it's actually true. His second campaign was a full-out invasion, however. But what were its goals? It seems very unlikely that Caesar intended to secure Britain, an island that was still pretty much entirely unknown, with such a force, especially when Gaul was still simmering at his back. It's difficult to tell what Caesar's real aims were here, because he covered his ass so thoroughly and insisted that his campaign had very limited goals. That may well be true, since his forces never really attempted the massive breakthroughs that Caesar's troops were famous for. But even so the campaign must have been somewhat disappointing. No territory was actually captured, even though on paper Caesar's goals of disrupting British supply of troops and money to the Gauls were accomplished. And valuable time was wasted, to very small gains.