(You've got some terminology confusion: the Chief Justice is the head of the court, the rest are just "justices")
When the Constitution was being written the framers were used to Great Britain's monarchy, and were somewhat eager to limit the monarch's (in the case of the U.S., the executive branch's) power. The U.S. government places a high value on checks and balances between the branches, and the English notion of the king arbitrarily firing a judge he didn't like was problematic. In the Federalist Papers #78, Hamilton wrote:
The standard of good behavior for the continuance in office of the Judicial magistracy, is certainly one of the most valuable of the modern improvements in the practice of Government. In a monarchy, it is an excellent barrier to the despotism of the Prince; in a republic it is a no less excellent barrier to the encroachments and oppressions of the representative body. And it is the best expedient which can be devised in any Government, to secure a steady, upright, and impartial administration of the laws.
In short, the court is around to make sure the other branches don't decide to start eliminating fundamental rights
Furthermore, term limits mean there's more pressure to keep the people happy; even if justices aren't directly elected like many other politicians, the opinion of the country will have a big impact on whether or not they keep their seats. From the same paper:
This independence of the Judges is equally requisite to guard the Constitution and the rights of individuals, from the effects of those ill humors, which the arts of designing men, or the influence of particular conjunctures, sometimes disseminate among the People themselves, and which, though they speedily give place to better information, and more deliberate reflection, have a tendency, in the mean time, to occasion dangerous innovations in the Government, and serious oppressions of the minor party in the community. Though I trust the friends of the proposed Constitution will never concur with its enemies, in questioning that fundamental principle of republican Government, which admits the right of the People to alter or abolish the established Constitution, whenever they find it inconsistent with their happiness, yet it is not to be inferred from this principle, that the Representatives of the People, whenever a momentary inclination happens to lay hold of a majority of their constituents, incompatible with the provisions in the existing Constitution, would, on that account, be justifiable in a violation of those provisions; or that the Courts would be under a greater obligation to connive at infractions in this shape, than when they had proceeded wholly from the cabals of the Representative body. Until the People have, by some solemn and authoritative act, annulled or changed the established form, it is binding upon themselves collectively, as well as individually; and no presumption, or even knowledge of their sentiments, can warrant their Representatives in a departure from it, prior to such an act. But it is easy to see, that it would require an uncommon portion of fortitude in the Judges to do their duty as faithful guardians of the Constitution, where Legislative invasions of it had been instigated by the major voice of the community.
Which I would interpret to mean: justices shouldn't need to fear for their jobs every time they make a decision that might be unpopular, even if it's in the country's interests long term, because most judges won't have the nerve to do what's right
As a sidenote, the notion of life appointments for Supreme Court justices hasn't been without controversy, particularly of late. Increased life expectencies means justices have been serving for longer than ever before, and some people feel that the checks against the judicial branch is considerably weaker than the other branches. Term Limits for the Supreme Court: Life Tenure Reconsidered proposed 18-year term limits, staggered so a new justice is appointed every two years. The Supreme Court Renewal Act proposed the same scheme