What was the purpose of the Varangian Guard? How and why did they form and why did they serve the Byzantine Empire?

In addition to the other sources named, Blöndal and Benedikz' The Varangians of Byzantium is worth a look. The book covers both the early and later history of the Guard, their use in warfare, their role in coups d'état, and of course the life of Harald Hardrada, who served in the Varangian Guard before becoming King of Norway and dying at the Battle of Stamford Bridge (popular myth has it that he joined the guard incognito, but probably everyone knew who he was). Blöndal comes from the unusual position of being able to read both old Norse and medieval Greek, though his understanding of Byzantine military organization and history is rather outdated. The work paints the Varangians as very close to the Emperor starting with the reign of Basil II (who organized the Guard): their leader, the akolouthos or "follower", would have been something like a secret service agent, following the Emperor in close proximity almost everywhere, and the Emperor would always have been flanked by his "axe-bearing barbarians" (pelekyphoroi barbaroi), possibly a reference to the so-called "Danish axe". The Varangians would also have saluted the ascendant Emperor in their own language during his inaugural parade. The role of the Varangians diminished substantially over time: for example, the akolouthos appears in lower and lower priority in processions.

One element of the Varangian "mythos" has it that they were a force for stability since, unlike native Rhomaioi, they were not interested in local political or religious intrigue, being loyal only to money and therefore to the office of emperor. Anna Komnena paints this picture in the Alexiad: upon the death of Ioannes Tzimiskes, whom the Varangians had failed to protect, they immediately knelt before his successor and murderer, Nikephoros II: "alive, they would have defended him to the last breath: dead, there was no point in avenging him". (Note that this episode, in 969, predates the organization of the Guard per se around 989, but Varangians were sometimes used as mercenaries or guards prior to Basil II. It illustrates the common myth surrounding the Varangians' unshakeable loyalty, though.) A few years ago I took a look at a Ph.D. thesis that challenged this idea, instead portraying the Varangians as mercurial, actively taking part in coups and palace intrigue, oftentimes to the point of fighting other Varangians who supported a different claimant to the throne: in this sense they made Byzantine succession bloodier and more unstable. I wish I could find it again.

As for why they served, it was common for Scandinavians to make the trek down the "Varangian Way" to the Black Sea for trade or raiding, or to seek fortune as an adventurer. Joining the Guard was one way to accomplish this. Blöndal claims that the Varangians were generally paid much better than other elements of the Byzantine military. Their "wage" would have been scaled against an entrance fee (higher entrance fee, higher wage), with recruits who could not pay the entrance fee taking a loan from the treasury. They would also have gathered substantial wealth from looting and plundering: on some campaigns, booty was divided so that one third went to the emperor, one third went to the Varangians, and one third went to the rest of the company. The thirty or so "Greece runestones" (Scandinavian runestones that attest to their subjects' success in "Greece" or "Miklagarðr", the old Norse name for Constantinople) provide evidence that serving in the Guard was a way to earn prestige, as well.